# Confidence Intervals in Excel

Save this PDF as:

Size: px
Start display at page:

## Transcription

1

2 Confidence Intervals in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. ISBN: Copyright Page 1

3 Table of Contents (Click On Chapters To Go To Them) Confidence Intervals in Excel 4 Basic Explanation of Confidence Interval 6 Mean Sampling vs. Proportion Sampling...6 Confidence Interval of a Population Mean 7 Calculate Confidence Intervals Using Large Samples.7 The Central Limit Theorem... 8 Levels of Confidence and Significance... 8 Population Mean vs. Sample Mean... 9 Standard Deviation and Standard Error... 9 Standard Deviation and Standard Error... 9 Region of Certainty vs. Region of Uncertainty Z Score Excel Functions Used When Calculating Confidence Interval of Mean Formulas for Calculating Confidence Interval Boundaries from Sample Data Problem 1: Calculate a Confidence Interval from a Random Sample of Test Scores.16 Problem 2: Calculate Confidence Interval Based on Sample Mean & Standard Deviation 19 Problem 3: Calculate Range of 95% of Sales Based on Population Mean & Stand. Dev 21 Determine Min Sample Size To Keep Confidence Interval of Mean Within Tolerance 23 Problem 4: Determine Min Number of Samples Needed To Limit Size of 95% Conf. Int 24 Confidence Interval of Population Proportion.. 25 Mean Sampling vs. Proportion Sampling Levels of Confidence and Significance Copyright Page 2

4 Population Proportion vs. Sample Proportion Standard Deviation and Standard Error Region of Certainty vs. Region of Uncertainty Z Score Excel Functions For Calculating Proportion Confidence Intervals Formulas for Calculating Proportion Confidence Intervals Formulas For Calculating Proportion Confidence Intervals...28 Problem 5: Calculate Confidence Interval For a Proportion of Shoppers Who Prefer To Pay By Credit Card..29 Min Sample Size To Keep Proportion Confidence Interval Within a Tolerance 31 Problem 6: Determine Min Sample Size of Voters To Obtain a 95% Confidence Interval Within 1% Certainty.32 Copyright Page 3

5 Confidence Intervals in Excel Confidence Intervals are estimates of a population's average or proportion based upon sample data drawn from the population. A Confidence Interval is a range of values in which the mean is likely to fall with a specified level of confidence or certainty. Copyright Page 4

6 Contents Basic Explanation of Confidence Intervals Mean Sampling vs. Proportion Sampling Confidence Intervals of a Population Mean Calculate Confidence Intervals Using Large Samples The Central Limit Theorem Levels of Confidence and Significance Population Mean vs. Sample Mean Standard Deviation and Standard Error Region of Certainty vs. Region of Uncertainty Z Score Excel Functions Used When Calculating Confidence Interval of Mean COUNT (Highlighted Block of Cells) STDEV (Highlighted Block of Cells) AVERAGE (Highlighted Block of Cells) NORMSINV (1 - α/2) CONFIDENCE (α, s, n) Formula for Calculating Confidence Interval Boundaries from Sample Data Problem 1: Calculate a Confidence Interval from a Random Sample of Test Scores Problem 2: Calculate a Confidence Interval of Daily Sales Based Upon Sample Mean and Standard Deviation Problem 3: Calculate an Exact Range of 95% of Sales Based Upon the Population Mean and Standard Deviation Determine Minimum Sample Size to Limit Confidence Interval of Mean to a Certain Width Problem 4: Determine the Minimum Number of Sales Territories to Sample In Order To Limit the 95% Confidence Interval to a Certain Width Confidence Interval of a Population Proportion Mean Sampling vs. Proportion Sampling Levels of Confidence and Significance Population Proportion vs. Sample Proportion Standard Deviation and Standard Error Region of Certainty vs. Region of Uncertainty Z Score Excel Functions Used When Calculating Confidence Interval of Proportion COUNT (Highlighted Block of Cells) NORMSINV (1 - α) Formula for Calculating Confidence Interval Boundaries from Sample Data Problem 5: Determine Confidence Interval of Shoppers Who Prefer to Pay By Credit Card Based Upon Sample Data Determine Minimum Sample Size to Limit Confidence Interval of Proportion to a Certain Width Problem 6: Determine the Minimum Sample Size of Voters to be 95% Certain that the Population Proportion is only 1% Different than Sample Proportion Copyright Page 5

7 Basic Explanation of Confidence Intervals The Confidence Interval is an interval in which the true population mean or proportion probably lies based upon a much smaller random sample taken from that population. Confidence Intervals for means are calculated differently than Confidence Intervals for proportions. The first half of this course module will discuss calculating a Confidence Interval for a population mean. The second half will cover calculating a Confidence Interval for a population proportion. First we will briefly discuss the difference between sampling for mean and sampling for proportion: Mean Sampling vs. Proportion Sampling What determines whether a mean is being estimated or a proportion is being estimated is the number of possible outcomes of each sample taken. Proportion samples have only two possible outcomes. For example, if you are comparing the proportion of Republicans in two different cities, each sample has only two possible values; the person sampled either is a Republican or is not. Mean samples have multiple possible outcomes. For example, if you are comparing the mean age of people in two different cities, each sample can have numerous values; the person sampled could be anywhere from 1 to 110 years old. Below is a description of how to calculate a Confidence Interval for a population's mean. Note that everything is almost the same as the calculation of the Confidence Interval for a population proportion, except sample standard error. Copyright Page 6

8 Confidence Interval of a Population Mean The Confidence Interval of a Mean is an interval in which the true population mean probably lies based upon a much smaller random sample taken from that population. A 95% Confidence Interval of a Mean is the interval that has a 95% chance of containing the true population mean. The width of a Confidence Interval is affected by the sample size. The larger the sample size, the more accurate and tighter is the estimate of the true population mean. The larger the sample size, the smaller will be the Confidence Interval. Samples taken must be random and also be representative of the population. Calculate Confidence Intervals Using Large Samples (n>30) Confidence Intervals are usually calculated and plotted on a Normal curve. If the sample size is less than 30, the population must be known to be Normally distributed. If small-sample data (n<30) is used to plot the Confidence Interval of the Mean for a population that is not Normally distributed, the result can be totally wrong. Probably the most common major mistake in statistics is to apply Normal or t-distribution tests to small-sample data taken from a population of unknown distribution. Typically the actual distribution of a population is not known. If the population's underlying distribution is not known (usually it is not), then only large samples (n>30) are valid for creating a Confidence Interval of the Mean. The most important theorem of statistics, the Central Limit Theorem, explains the reason for this. Copyright Page 7

9 The Central Limit Theorem The Central Limit Theorem is statistics' most fundamental theorem. In a nutshell, it states the following: Random sample data can be plotted on a Normal curve to estimate a population's mean no matter how the population is distributed, as long as sample size is large (n>30). The above definition of the Central Limit Theorem is the most practical and easy-to-understand. The following definition of this theorem is a bit more technical and will satisfy statisticians (but basically says the same thing as the above): No matter how the population is distributed, the sampling distribution of the mean approaches the Normal curve as sample size becomes large. Levels of Confidence and Significance Level of Significance, α ("alpha"), equals the maximum allowed percent of error. If the maximum allowed error is 5%, then α = Level of Confidence is the desired degree of certainty. A 95% Confidence Level is the most common. A 95% Confidence Level would correspond to a 95% Confidence Interval of the Mean. This would state that the actual population mean has a 95% probability of lying within the calculated interval. A 95% Confidence Level corresponds to a 5% Level of Significance, or α = The Confidence Level therefore equals 1 - α. Copyright Page 8

10 Population Mean vs. Sample Mean Population Mean = µ ("mu") (This is what we are trying to estimate) Sample Mean = x avg Standard Deviation and Standard Error Standard Deviation is a measure of statistical dispersion. It's formula is the following: SQRT ( [ SUM (x - x avg )^2 ] / N ). There is no need to memorize the formula because you can plug in Excel's STDEV function discussed later. Standard Deviation equals the square root of the Variance. Population Standard Deviation = σ ("sigma") Sample Standard Deviation = s Standard Error is an estimate of population Standard Deviation from data taken from a sample. If the population Standard Deviation, σ, is known, then the Sample Standard Error, s xavg, can be calculated. If only the Sample Standard Deviation, s, is known, then Sample Standard Error, s xavg, can be estimated by substituting Sample Standard Deviation, s, for Population Standard Deviation, σ, as follows: Sample Standard Error = s xavg = σ / SQRT(n) s / SQRT(n) σ = Population standard deviation s = Sample standard deviation n = sample size Copyright Page 9

11 Region of Certainty vs. Region of Uncertainty Region of Certainty is the area under the Normal curve that corresponds to the required Level of Confidence. If a 95% percent Level of Confidence is required, then the Region of Certainty will contain 95% of the area under the Normal curve. The outer boundaries of the Region of Certainty will be the outer boundaries of the Confidence Interval. The Region of Certainty, and therefore the Confidence Interval, will be centered about the mean. Half of the Confidence Interval is on one side of the mean and half on the other side. Region of Uncertainty is the area under the Normal curve that is outside of the Region of Certainty. Half of the Region of Uncertainty will exist in the right outer tail of the Normal curve and the other half in the left outer tail. This is similar to the concept of the "two-tailed test" that is used in Hypothesis testing in further sections of this course. The concepts of one- and two-tailed testing are not used when calculating Confidence Intervals. Just remember that the Region of Certainty, and therefore the Confidence Interval, are always centered about the mean on the Normal curve. Relationship Between Region of Certainty, Uncertainty, and Alpha The Region of Uncertainty corresponds to α ("alpha"). If α = 0.05, then that Region of Uncertainty contains 5% of the area under the Normal curve. Half of that area (2.5%) is in each outer tail. The 95% area centered about the mean will be the Region of Certainty. The outer boundaries of this Region of Certainty will be the outer boundaries of the 95% Confidence Interval. The Level of Confidence is 95% and the Level of Significance, or maximum error allowed, is 5%. Copyright Page 10

12 Illustrating a two-tail test Similar to what is used for calculating Confidence Interval Copyright Page 11

13 Illustrating a one-tailed test Right Tail Illustrating a one-tailed test Left Tail Copyright Page 12

14 Z Score Z Score is the number of Standard Errors from the mean to outer right boundary of the Region of Certainty (and therefore to the outer right boundary of the Confidence Interval). Standard Errors are used and not Standard Deviations because sample data is being used to calculate the Confidence Interval. Z Score is calculated by the following Excel function: Z Score (1-α) = NORMSINV (1 - α/2) - This will be discussed below. It is very important to note that on a Standardized Normal Curve, the Distance from the mean to boundary of the Region of Certainty equals the number of standard errors from the mean to boundary, which is the Z Score. The above is only true for a Standardized Normal Curve. It is NOT true for a Non-Standardized Normal curve. Copyright Page 13

15 Excel Functions Used When Calculating Confidence Interval of Mean COUNT (Highlighted block of cells) = Sample size = n ----> Counts number of cells in highlighted block STDEV (Highlighted block of cells) = Standard deviation ----> Calculates Standard Deviation of all cells in highlighted block AVERAGE (Highlighted block of cells) = Mean ----> Calculates the mean of all cells in highlighted block NORMSINV (1 - α/2) = Z Score (1 - α) = Number of Standard errors from mean to boundary of Confidence Interval. Note that (1 - α/2) = the entire area in the Normal curve to the left of outer right boundary of the Region of Certainty, or Confidence Interval. This includes the entire Region of Certainty and the half of the Region of Uncertainty that exists in the left tail. For example: Level of Confidence = 95% for a 95% Confidence Interval Level of Significance = 5% (α = 0.05) 1 - α = 0.95 = 95% Z Score 95% = NORMSINV(1 α/2) = NORMSINV (1 -.05/2) = NORMSINV( ) Z Score 95% = NORMSINV (0.975) = 1.96 The outer right boundary of the 95% Confidence Interval, and the Region of Certainty, is 1.96 Standard Errors from the mean. The left boundary is the same distance from the mean because the Confidence Interval is centered about the mean. CONFIDENCE ( α, s, n ) = Width of half of the Confidence Interval α = Level of Significance s = Sample Standard Deviation - Note that this is not Standard Error. s is calculated by applying STDEV to the sample values. n = Sample size - Apply COUNT to sample values. Copyright Page 14

16 Formulas for Calculating Confidence Interval Boundaries from Sample Data Confidence Interval Boundaries = Sample mean +/- Z Score (1-α) * Sample Standard Error Confidence Interval Boundaries = x avg +/- Z Score (1-α) * s xavg Sample Mean = x avg = AVERAGE (Highlighted block of cell containing samples) Z Score (1 - α) = NORMSINV (1 - α/2) Sample Standard Error = s xavg = σ / SQRT(n) s / SQRT(n) Sample size = n = COUNT (Highlighted block of cells containing samples) Sample Standard Deviation = s = STDEV (Highlighted block of cells containing samples) CONFIDENCE ( α, s, n ) = Width of half of the Confidence Interval CONFIDENCE ( α, s, n ) = Z Score (1-α) * s xavg So: Confidence Interval Boundaries = x avg +/- Z Score (1-α) * s xavg Confidence Interval Boundaries = x avg +/- CONFIDENCE ( α, s, n ) Copyright Page 15

17 Problem 1: Calculate a Confidence Interval from a Random Sample of Test Scores Problem: Given the following set of 32 random test scores taken from a much larger population, calculate with 95% certainty an interval in which the population mean test score must fall. In other words, calculate the 95% Confidence Interval for the population test score mean. The random sample of 32 tests scores is shown on the next page. Copyright Page 16

18 Copyright Page 17

19 Copyright Page 18

20 Problem 2: Calculate a Confidence Interval of Daily Sales Based Upon Sample Mean and Standard Deviation Problem: Average daily demand for books sold in a small Barnes and Noble store is 455 books with a standard deviation of 200. This average and standard deviation are taken from sale data collected every day for a period of 60 days. What is the range that the true average daily book sales lies in with 95% certainty? Level of Confidence = 95% = 1 - α Level of Significance = α = 0.05 Sample Size = n = 60 Sample Mean = x avg = 455 Sample Standard Deviation = s = 200 Sample Standard Error = s xavg = σ / SQRT(n) s / SQRT(n) s xavg = σ / SQRT(n) 200 / SQRT(60) = 25.8 Z Score (1 - α) = Z Score 95% = NORMSINV (1 - α/2) = NORMSINV ( ) = NORMSINV (0.975) = 1.96 Width of Half the Confidence Interval = CONFIDENCE ( α, s, n) = CONFIDENCE ( 0.05, 200, 60) = 50.6 Also, equivalently: Width of Half the Confidence Interval = Z Score (1-α) * s xavg = 1.96 * 25.8 = 50.6 Confidence Interval Boundaries = x avg +/- Z Score (1-α) * s xavg = 455 +/- (1.96)*(25.8) = 455 +/ = to Copyright Page 19

21 Confidence Interval Boundaries = x avg +/- CONFIDENCE ( α, s, n) = 455 +/ = to Copyright Page 20

22 Problem 3: Calculate an Exact Range of 95% of Sales Based Upon the Upon the Population Mean and Standard Deviation Problem: Average daily demand for books sold in a large Barnes and Noble store is 5,000 books with a standard deviation of 200. This average and standard deviation are taken from sale data collected every day for a period of 5 years. What is the range that 95% of the daily unit book sales fall in? The daily sales data is Normally distributed. This problem is not a Confidence Interval problem. We do not need to create an estimate of the population mean (a Confidence Interval) because we know exactly what it is. We are given the population mean and population standard deviation. We do need to know how the population is distributed in order to calculate the interval that contains 95% of all population data. Given that the population is Normally distributed, we simply need to map the region of this Normal curve that contains 95% of the total area and is centered about the mean as follows: Population mean = µ = 5,000 Population Standard Deviation = σ = 200 Range Containing 95% of Sales Data = µ +/- [ Z Score 95% * σ ] = 5,000 +/- [ NORMSINV(0.975) * 200 ] = 5,000 +/- 392 = 4,608 to 5,392 Copyright Page 21

23 Copyright Page 22

24 Determining Minimum Sample Size (n) to Keep Confidence Interval of the Mean within a Certain Tolerance The larger the sample size, the more accurate and tighter will be the prediction of a population's mean. Stated another way, the larger the sample size, the smaller will be the Confidence Interval of the population's mean. Width of the Confidence Interval is reduced when sample size is increased. Quite often a population's mean needs to be estimated with some level of certainty to within plus or minus a specified tolerance. This specified tolerance is half the width of the Confidence Interval. Sample size directly affects the width of the Confidence Interval. The relationship between sample size and width of the Confidence Interval is shown as follows: CONFIDENCE ( α, s, n ) = Width of half of the Confidence Interval CONFIDENCE ( α, s, n ) = Z Score (1-α) * s xavg Width of Half of the Confidence Interval = Z Score (1-α) * s xavg Width of Half of the Confidence Interval = Z Score (1-α) * σ / SQRT(n) With algebraic manipulation of the above we have: Z Score (1-α) * s / SQRT(n) SQRT(n) = Z Score (1-α) * σ / Width of Half of the Confidence Interval n = [ Z Score (1-α) ] 2 * [σ] 2 / [ Width of Half of the Confidence Interval ] 2 Also, if we only have sample standard deviation, s, and not population standard deviation, σ : n [ Z Score (1-α) ] 2 * [s] 2 / [ Width of Half of the Confidence Interval ] 2 Copyright Page 23

25 Problem 4: Determine the Minimum Number of Sales Territories to Sample In Order To Limit the 95% Confidence Interval to a Certain Width Problem: A national sales manager in charge of 5,000 similar territories ran a nationwide promotion. He then collected sales data from a random sample of the territories to evaluate sales increase. From the sample, the average sales increase per territory was \$10,000 with a standard deviation of \$500. How many territories would he have had to have sampled to be 95% sure that the actual nationwide average territory sales increase was no more than \$50 different than average territory sales increase from the sample he took? Level of Confidence = 95% = 1 - α Level of Significance = α = 0.05 Sample Size = n =? Sample Mean = x avg > Note this does not need to be known to solve this problem Sample Standard Deviation = s = 500 Z Score (1 - α) = Z Score 95% = NORMSINV (1 - α/2) Width of Half the Confidence Interval = 50 = NORMSINV ( ) = NORMSINV (0.975) = 1.96 n [ Z Score (1-α) ] 2 * [s] 2 / [ Width of Half of the Confidence Interval ] 2 n [ 1.96 ] 2 * [500] 2 / [ 50 ] 2 = 384 The sales manager would have to sample at least 384 territories to be 95% certain that nationwide territory average was within +/- \$50 of the sample territory average. Note that the 95% confidence interval is \$10,000 +/- \$50 and this interval has a width = \$100 if sample size is 384. ****************************************************************************************************************************************** Copyright Page 24

26 Confidence Interval of a Population Proportion Creating a Confidence Interval for a population's proportion is very similar to creating a Confidence Interval for a population's mean. The only real difference is how the standard error is calculated. Everything else is the same. The method of calculating a Confidence Interval for a population mean was covered in detail earlier in this module. First, the difference between using sampling to estimate a population mean and using sampling to estimate a population proportion will be explained below: Mean Sampling vs. Proportion Sampling What determines whether a mean is being estimated or a proportion is being estimated is the number of possible outcomes of each sample taken. Proportion samples have only two possible outcomes. For example, if you are comparing the proportion of Republicans in two different cities, each sample has only two possible values; the person sampled either is a Republican or is not. Mean samples have multiple possible outcomes. For example, if you are comparing the mean age of people in two different cities, each sample can have numerous values; the person sampled could be anywhere from 1 to 110 years old. Below is a description of how to calculate a Confidence Interval for a population's proportion. Note that everything is almost the same as the calculation of the Confidence Interval for a mean, except sample standard error. Copyright Page 25

27 Levels of Confidence and Significance Level of Significance, α ("alpha"), equals the maximum allowed percent of error. If the maximum allowed error is 5%, then α = Level of Confidence is selected by the user. A 95% Level is the most common. A 95% Confidence Level would correspond to a 95% Confidence Interval of the Proportion. This would state that the actual population Proportion has a 95% probability of lying within the calculated interval. A 95% Confidence Level corresponds to a 5% Level of Significance, or α = The Confidence Level therefore equals 1 - α. Population Proportion vs. Sample Proportion Population Proportion = µ p = p (This is what we are trying to estimate) Sample Proportion = p avg Standard Deviation and Standard Error Standard Deviation is not calculated during the creation of Confidence Interval for a population proportion. Standard Error is an estimate of population Standard Deviation from data taken from a sample. Sample Standard Error will be an estimate taken from the sample proportion, p avg, and sample size, n. This is the major difference between calculating a Confidence Interval for a proportion and for a mean. Binomial distribution rules apply to proportions because a proportion sample has only two possible outcomes, just like a binomial variable. Sample Standard Error of a Proportion = σ pavg = SQRT(p * q / n) s pavg Estimated Sample Standard Error of a Proportion = s pavg = SQRT ( p avg * q avg / n ) p = Population proportion - This is the unknown that will be estimated with a Confidence Interval q = 1 - p n = sample size p avg = Sample proportion q avg = 1 - p avg Copyright Page 26

28 Region of Certainty vs. Region of Uncertainty Region of Certainty is the area under the Normal curve that corresponds to the required Level of Confidence. If a 95% percent Level of Confidence is required, then the Region of Certainty will contain 95% of the area under the Normal curve. The outer boundaries of the Region of Certainty will be the outer boundaries of the Confidence Interval. The Region of Certainty, and therefore the Confidence Interval, will be centered about the mean. Half of the Confidence Interval is on one side of the mean and half on the other side. Region of Uncertainty is the area under the Normal curve that is outside of the Region of Certainty. Half of the Region of Uncertainty will exist in the right outer tail of the Normal curve and the other half in the left outer tail. This is similar to the concept of the "two-tailed test" that is used in Hypothesis testing in further sections of this course. The concepts of one and two-tailed testing are not used when calculating Confidence Intervals. Just remember that the Region of Certainty, and therefore the Confidence Interval, are always centered about the mean on the Normal curve. Relationship Between Region of Certainty, Uncertainty, and Alpha The Region of Uncertainty corresponds to α ("alpha"). If α = 0.05, then that Region of Uncertainty contains 5% of the area under the Normal curve. Half of that area (2.5%) is in each outer tail. The 95% area centered about the mean will be the Region of Certainty. The outer boundaries of this Region of Certainty will be the outer boundaries of the 95% Confidence Interval. The Level of Confidence is 95% and the Level of Significance, or maximum error allowed, is 5%. Z Score Z Score is the number of Standard Errors from the mean to outer right boundary of the Region of Certainty (and therefore to the outer right boundary of the Confidence Interval). Standard Errors are used and not Standard Deviations because sample data is being used to calculate the Confidence Interval. Z Score is calculated by the following Excel function: Z Score (1-α) = NORMSINV (1 - α/2) - This will be discussed shortly. Copyright Page 27

29 Excel Functions Used When Calculating Confidence Interval for a Population Proportion Note that Excel functions STDEV and AVERAGE are not used when working with proportions. The CONFIDENCE function is not used either. COUNT (Highlighted block of cells) = Sample size = n ----> Counts number of cells in highlighted block NORMSINV (1 - α/2) = Z Score (1 - α) = Number of Standard Errors from mean to boundary of Confidence Interval. Note that (1 - α/2) = the entire area in the Normal curve to the left of outer right boundary of the Region of Certainty, or Confidence Interval. This includes the entire Region of Certainty and the half of the Region of Uncertainty that exists in the left tail. For example: Level of Confidence = 95% for a 95% Confidence Interval Level of Significance = 5% (α = 0.05) 1 - α = 0.95 = 95% Z Score 95% = NORMSINV (1 - α/2) = NORMSINV (1 -.05/2) = NORMSINV( ) Z Score 95% = NORMSINV (0.975) = 1.96 The outer right boundary of the 95% Confidence Interval, and the Region of Certainty, is 1.96 Standard Errors from the mean. The left boundary is the same distance from the mean because the Confidence Interval is centered about the mean. Formula for Calculating Confidence Interval Boundaries from Sample Data for a Population Proportion Confidence Interval Boundaries = Sample proportion +/- Z Score (1-α) * Sample Standard Error Confidence Interval Boundaries = p avg +/- Z Score (1-α) * s pavg Sample Proportion = p avg Z Score (1 - α) = NORMSINV (1 - α/2) Sample Standard Error of a Proportion = σ pavg s pavg = SQRT ( p avg * q avg / n ) Sample size = n = COUNT (Highlighted block of cells containing samples) Confidence Interval Boundaries = p avg +/- Z Score (1-α) * s pavg Copyright Page 28

30 Problem 5: Determine Confidence Interval of Shoppers Who Prefer to Pay By Credit Card Based Upon Sample Data Problem: A random sample of 1,000 shoppers was taken. 70% preferred to pay with a credit card. 30% preferred to pay with cash. Determine the 95% Confidence Interval for the proportion of the general population that prefers to pay with a credit card. Level of Confidence = 95% = 1 - α Level of Significance = α = 0.05 Sample Size = n = 1,000 Sample Proportion = p avg = 0.70 q avg = 1 - p avg = 0.30 Sample Standard Error of a Proportion = σ pavg s pavg = SQRT ( p avg * q avg / n ) s pavg = SQRT ( 0.70 * 0.30 / 1,000 ) = Z Score (1 - α) = Z Score 95% = NORMSINV (1 - α/2) = NORMSINV ( ) = NORMSINV (0.975) = 1.96 Width of Half the Confidence Interval = Z Score (1-α) * s pavg = 1.96 * = Confidence Interval Boundaries = p avg +/- Z Score (1-α) * s pavg = /- (1.96) * (0.014) = / = to = 67.26% to 72.74% Copyright Page 29

31 Copyright Page 30

32 Determining Minimum Sample Size (n) to Keep Confidence Interval of the Proportion within a Certain Tolerance The larger the sample size, the more accurate and tighter will be the prediction of a population's mean. Stated another way, the larger the sample size, the smaller will be the Confidence Interval of the population's mean. Width of the Confidence Interval is reduced when sample size is increased. Quite often a population's mean needs to be estimated with some level of certainty to within plus or minus a specified tolerance. This specified tolerance is half the width of the confidence interval. Sample size directly affects the width of the Confidence Interval. The relationship between sample size and width of the Confidence Interval is shown as follows: Width of Half the Confidence Interval = Z Score (1-α) * s pavg s pavg = SQRT ( p avg * q avg / n ) Width of Half the Confidence Interval = Z Score (1-α) * SQRT ( p avg * q avg / n ) [ Width of Half the Confidence Interval ] 2 = [ Z Score (1-α) ] 2 * ( p avg * q avg / n ) n = [ Z Score (1-α) ] 2 * ( p avg * q avg ) / [ Width of Half the Confidence Interval ] 2 Copyright Page 31

33 Problem 6: Determine the Minimum Sample Size of Voters to be 95% Certain that the Population Proportion is no more than 1% Different from Sample Proportion Problem: A random survey was conducted in one city to learn voting preferences. 40% of voters surveyed said they would vote Republican. 60% of the voters surveyed said they would vote Democrat. Determine the minimum number of voters that had to be surveyed to be 95% certain that the results were accurate within +/- 1%. Level of Confidence = 95% = 1 - α Level of Significance = α = 0.05 p avg = 0.40 q avg = 1 - p avg = 0.60 Width of Half the Confidence Interval = > (1%) Z Score (1 - α) = Z Score 95% = NORMSINV (1 - α/2) = NORMSINV ( ) = NORMSINV (0.975) = 1.96 n = [ Z Score (1-α) ] 2 * ( p avg * q avg ) / [ Width of Half the Confidence Interval ] 2 n = [ 1.96 ] 2 * ( 0.40 * 0.60 ) / [ 0.01 ] 2 = 9,220 At least 9,220 random voters had to be surveyed to be 95% certain that the population proportion is no more than 1% different from the sample. Copyright Page 32

34 Meet the Author Mark Harmon is a master number cruncher. Creating overloaded Excel spreadsheets loaded with complicated statistical analysis is his idea of a good time. His profession as an Internet marketing manager provides him with the opportunity and the need to perform plenty of meaningful statistical analysis at his job. Mark Harmon is also a natural teacher. As an adjunct professor, he spent five years teaching more than thirty semester-long courses in marketing and finance at the Anglo-American College in Prague and the International University in Vienna, Austria. During that five-year time period, he also worked as an independent marketing consultant in the Czech Republic and performed longterm assignments for more than one hundred clients. His years of teaching and consulting have honed his ability to present difficult subject matter in an easy-tounderstand way. Harmon received a degree in electrical engineering from Villanova University and MBA in marketing from the Wharton School. Copyright Page 33

35

### Normality Testing in Excel

Normality Testing in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com

### t Tests in Excel The Excel Statistical Master By Mark Harmon Copyright 2011 Mark Harmon

t-tests in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com www.excelmasterseries.com

### Statistical Inference

Statistical Inference Idea: Estimate parameters of the population distribution using data. How: Use the sampling distribution of sample statistics and methods based on what would happen if we used this

### Estimation of the Mean and Proportion

1 Excel Manual Estimation of the Mean and Proportion Chapter 8 While the spreadsheet setups described in this guide may seem to be getting more complicated, once they are created (and tested!), they will

### Hypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam

Hypothesis Testing 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 3 2. Hypothesis Testing... 3 3. Hypothesis Tests Concerning the Mean... 10 4. Hypothesis Tests

### Standard Deviation Calculator

CSS.com Chapter 35 Standard Deviation Calculator Introduction The is a tool to calculate the standard deviation from the data, the standard error, the range, percentiles, the COV, confidence limits, or

### Constructing and Interpreting Confidence Intervals

Constructing and Interpreting Confidence Intervals Confidence Intervals In this power point, you will learn: Why confidence intervals are important in evaluation research How to interpret a confidence

### HYPOTHESIS TESTING: POWER OF THE TEST

HYPOTHESIS TESTING: POWER OF THE TEST The first 6 steps of the 9-step test of hypothesis are called "the test". These steps are not dependent on the observed data values. When planning a research project,

### HypoTesting. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.

Name: Class: Date: HypoTesting Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A Type II error is committed if we make: a. a correct decision when the

### Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means

Lesson : Comparison of Population Means Part c: Comparison of Two- Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis

### Basic Statistics. Probability and Confidence Intervals

Basic Statistics Probability and Confidence Intervals Probability and Confidence Intervals Learning Intentions Today we will understand: Interpreting the meaning of a confidence interval Calculating the

### LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.

### CALCULATIONS & STATISTICS

CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents

### Sampling Distribution of a Sample Proportion

Sampling Distribution of a Sample Proportion From earlier material remember that if X is the count of successes in a sample of n trials of a binomial random variable then the proportion of success is given

### CHAPTER 11 SECTION 2: INTRODUCTION TO HYPOTHESIS TESTING

CHAPTER 11 SECTION 2: INTRODUCTION TO HYPOTHESIS TESTING MULTIPLE CHOICE 56. In testing the hypotheses H 0 : µ = 50 vs. H 1 : µ 50, the following information is known: n = 64, = 53.5, and σ = 10. The standardized

### 4. Introduction to Statistics

Statistics for Engineers 4-1 4. Introduction to Statistics Descriptive Statistics Types of data A variate or random variable is a quantity or attribute whose value may vary from one unit of investigation

### 4. Continuous Random Variables, the Pareto and Normal Distributions

4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random

### 1 SAMPLE SIGN TEST. Non-Parametric Univariate Tests: 1 Sample Sign Test 1. A non-parametric equivalent of the 1 SAMPLE T-TEST.

Non-Parametric Univariate Tests: 1 Sample Sign Test 1 1 SAMPLE SIGN TEST A non-parametric equivalent of the 1 SAMPLE T-TEST. ASSUMPTIONS: Data is non-normally distributed, even after log transforming.

### Confidence intervals

Confidence intervals Today, we re going to start talking about confidence intervals. We use confidence intervals as a tool in inferential statistics. What this means is that given some sample statistics,

### Hypothesis Testing Summary

Hypothesis Testing Summary Hypothesis testing begins with the drawing of a sample and calculating its characteristics (aka, statistics ). A statistical test (a specific form of a hypothesis test) is an

### 7 Hypothesis testing - one sample tests

7 Hypothesis testing - one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X

### Section 7.1. Introduction to Hypothesis Testing. Schrodinger s cat quantum mechanics thought experiment (1935)

Section 7.1 Introduction to Hypothesis Testing Schrodinger s cat quantum mechanics thought experiment (1935) Statistical Hypotheses A statistical hypothesis is a claim about a population. Null hypothesis

### Introduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures.

Introduction to Hypothesis Testing Point estimation and confidence intervals are useful statistical inference procedures. Another type of inference is used frequently used concerns tests of hypotheses.

### PASS Sample Size Software

Chapter 250 Introduction The Chi-square test is often used to test whether sets of frequencies or proportions follow certain patterns. The two most common instances are tests of goodness of fit using multinomial

### Simple Regression Theory II 2010 Samuel L. Baker

SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

### 4: Probability. What is probability? Random variables (RVs)

4: Probability b binomial µ expected value [parameter] n number of trials [parameter] N normal p probability of success [parameter] pdf probability density function pmf probability mass function RV random

### Hypothesis Testing for Beginners

Hypothesis Testing for Beginners Michele Piffer LSE August, 2011 Michele Piffer (LSE) Hypothesis Testing for Beginners August, 2011 1 / 53 One year ago a friend asked me to put down some easy-to-read notes

### HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men

### Power and Sample Size Determination

Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 Power 1 / 31 Experimental Design To this point in the semester,

### AP * Statistics Review

AP * Statistics Review Confidence Intervals Teacher Packet AP* is a trademark of the College Entrance Examination Board. The College Entrance Examination Board was not involved in the production of this

### THE FIRST SET OF EXAMPLES USE SUMMARY DATA... EXAMPLE 7.2, PAGE 227 DESCRIBES A PROBLEM AND A HYPOTHESIS TEST IS PERFORMED IN EXAMPLE 7.

THERE ARE TWO WAYS TO DO HYPOTHESIS TESTING WITH STATCRUNCH: WITH SUMMARY DATA (AS IN EXAMPLE 7.17, PAGE 236, IN ROSNER); WITH THE ORIGINAL DATA (AS IN EXAMPLE 8.5, PAGE 301 IN ROSNER THAT USES DATA FROM

### Chapter Additional: Standard Deviation and Chi- Square

Chapter Additional: Standard Deviation and Chi- Square Chapter Outline: 6.4 Confidence Intervals for the Standard Deviation 7.5 Hypothesis testing for Standard Deviation Section 6.4 Objectives Interpret

### Factorial Analysis of Variance

Chapter 560 Factorial Analysis of Variance Introduction A common task in research is to compare the average response across levels of one or more factor variables. Examples of factor variables are income

### Standard Deviation Estimator

CSS.com Chapter 905 Standard Deviation Estimator Introduction Even though it is not of primary interest, an estimate of the standard deviation (SD) is needed when calculating the power or sample size of

### Confidence Intervals for One Standard Deviation Using Standard Deviation

Chapter 640 Confidence Intervals for One Standard Deviation Using Standard Deviation Introduction This routine calculates the sample size necessary to achieve a specified interval width or distance from

### HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men

### Prob & Stats. Chapter 9 Review

Chapter 9 Review Construct the indicated confidence interval for the difference between the two population means. Assume that the two samples are independent simple random samples selected from normally

### Hypothesis Testing or How to Decide to Decide Edpsy 580

Hypothesis Testing or How to Decide to Decide Edpsy 580 Carolyn J. Anderson Department of Educational Psychology University of Illinois at Urbana-Champaign Hypothesis Testing or How to Decide to Decide

### Hypothesis Testing. Bluman Chapter 8

CHAPTER 8 Learning Objectives C H A P T E R E I G H T Hypothesis Testing 1 Outline 8-1 Steps in Traditional Method 8-2 z Test for a Mean 8-3 t Test for a Mean 8-4 z Test for a Proportion 8-5 2 Test for

### AMS 5 CHANCE VARIABILITY

AMS 5 CHANCE VARIABILITY The Law of Averages When tossing a fair coin the chances of tails and heads are the same: 50% and 50%. So if the coin is tossed a large number of times, the number of heads and

### REGRESSION LINES IN STATA

REGRESSION LINES IN STATA THOMAS ELLIOTT 1. Introduction to Regression Regression analysis is about eploring linear relationships between a dependent variable and one or more independent variables. Regression

### Chapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing

Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing 8-3 Testing a Claim About a Proportion 8-5 Testing a Claim About a Mean: s Not Known 8-6 Testing

### An interval estimate (confidence interval) is an interval, or range of values, used to estimate a population parameter. For example 0.476<p<0.

Lecture #7 Chapter 7: Estimates and sample sizes In this chapter, we will learn an important technique of statistical inference to use sample statistics to estimate the value of an unknown population parameter.

### Measures of Central Tendency and Variability: Summarizing your Data for Others

Measures of Central Tendency and Variability: Summarizing your Data for Others 1 I. Measures of Central Tendency: -Allow us to summarize an entire data set with a single value (the midpoint). 1. Mode :

### 6.4 Normal Distribution

Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

### Bowerman, O'Connell, Aitken Schermer, & Adcock, Business Statistics in Practice, Canadian edition

Bowerman, O'Connell, Aitken Schermer, & Adcock, Business Statistics in Practice, Canadian edition Online Learning Centre Technology Step-by-Step - Excel Microsoft Excel is a spreadsheet software application

### UCLA STAT 13 Statistical Methods - Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates

UCLA STAT 13 Statistical Methods - Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates 1. (a) (i) µ µ (ii) σ σ n is exactly Normally distributed. (c) (i) is approximately Normally

### Null Hypothesis H 0. The null hypothesis (denoted by H 0

Hypothesis test In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing a claim about a property

### Statistical Significance and Bivariate Tests

Statistical Significance and Bivariate Tests BUS 735: Business Decision Making and Research 1 1.1 Goals Goals Specific goals: Re-familiarize ourselves with basic statistics ideas: sampling distributions,

### Module 7: Hypothesis Testing I Statistics (OA3102)

Module 7: Hypothesis Testing I Statistics (OA3102) Professor Ron Fricker Naval Postgraduate School Monterey, California Reading assignment: WM&S chapter 10.1-10.5 Revision: 2-12 1 Goals for this Module

### An Introduction to Sampling

An Introduction to Sampling Sampling is the process of selecting a subset of units from the population. We use sampling formulas to determine how many to select because it is based on the characteristics

### Statistical Inference and t-tests

1 Statistical Inference and t-tests Objectives Evaluate the difference between a sample mean and a target value using a one-sample t-test. Evaluate the difference between a sample mean and a target value

### Hypothesis Testing (unknown σ)

Hypothesis Testing (unknown σ) Business Statistics Recall: Plan for Today Null and Alternative Hypotheses Types of errors: type I, type II Types of correct decisions: type A, type B Level of Significance

### 5.1 Identifying the Target Parameter

University of California, Davis Department of Statistics Summer Session II Statistics 13 August 20, 2012 Date of latest update: August 20 Lecture 5: Estimation with Confidence intervals 5.1 Identifying

### Continuous Random Variables and the Normal Distribution

CHAPTER 6 Continuous Random Variables and the Normal Distribution CHAPTER OUTLINE 6.1 The Standard Normal Distribution 6.2 Standardizing a Normal Distribution 6.3 Applications of the Normal Distribution

### Outline. Correlation & Regression, III. Review. Relationship between r and regression

Outline Correlation & Regression, III 9.07 4/6/004 Relationship between correlation and regression, along with notes on the correlation coefficient Effect size, and the meaning of r Other kinds of correlation

### BA 275 Review Problems - Week 5 (10/23/06-10/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380-394

BA 275 Review Problems - Week 5 (10/23/06-10/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380-394 1. Does vigorous exercise affect concentration? In general, the time needed for people to complete

### Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY

Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY 1. Introduction Besides arriving at an appropriate expression of an average or consensus value for observations of a population, it is important to

### Six Sigma: Sample Size Determination and Simple Design of Experiments

Six Sigma: Sample Size Determination and Simple Design of Experiments Short Examples Series using Risk Simulator For more information please visit: www.realoptionsvaluation.com or contact us at: admin@realoptionsvaluation.com

### Module 5 Hypotheses Tests: Comparing Two Groups

Module 5 Hypotheses Tests: Comparing Two Groups Objective: In medical research, we often compare the outcomes between two groups of patients, namely exposed and unexposed groups. At the completion of this

### Histogram. Graphs, and measures of central tendency and spread. Alternative: density (or relative frequency ) plot /13/2004

Graphs, and measures of central tendency and spread 9.07 9/13/004 Histogram If discrete or categorical, bars don t touch. If continuous, can touch, should if there are lots of bins. Sum of bin heights

### Sampling and Hypothesis Testing

Population and sample Sampling and Hypothesis Testing Allin Cottrell Population : an entire set of objects or units of observation of one sort or another. Sample : subset of a population. Parameter versus

### Basic Quantum Mechanics Prof. Ajoy Ghatak Department of Physics. Indian Institute of Technology, Delhi

Basic Quantum Mechanics Prof. Ajoy Ghatak Department of Physics. Indian Institute of Technology, Delhi Module No. # 02 Simple Solutions of the 1 Dimensional Schrodinger Equation Lecture No. # 7. The Free

### Chapter 3: Nonparametric Tests

B. Weaver (15-Feb-00) Nonparametric Tests... 1 Chapter 3: Nonparametric Tests 3.1 Introduction Nonparametric, or distribution free tests are so-called because the assumptions underlying their use are fewer

### Chapter 7 Part 2. Hypothesis testing Power

Chapter 7 Part 2 Hypothesis testing Power November 6, 2008 All of the normal curves in this handout are sampling distributions Goal: To understand the process of hypothesis testing and the relationship

### Confidence Intervals for the Difference Between Two Means

Chapter 47 Confidence Intervals for the Difference Between Two Means Introduction This procedure calculates the sample size necessary to achieve a specified distance from the difference in sample means

### SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Ch. 10 Chi SquareTests and the F-Distribution 10.1 Goodness of Fit 1 Find Expected Frequencies Provide an appropriate response. 1) The frequency distribution shows the ages for a sample of 100 employees.

### Practice Exam. 1. What is the median of this data? A) 64 B) 63.5 C) 67.5 D) 59 E) 35

Practice Exam Use the following to answer questions 1-2: A census is done in a given region. Following are the populations of the towns in that particular region (in thousands): 35, 46, 52, 63, 64, 71,

### Chapter 9, Part A Hypothesis Tests. Learning objectives

Chapter 9, Part A Hypothesis Tests Slide 1 Learning objectives 1. Understand how to develop Null and Alternative Hypotheses 2. Understand Type I and Type II Errors 3. Able to do hypothesis test about population

### MBA 611 STATISTICS AND QUANTITATIVE METHODS

MBA 611 STATISTICS AND QUANTITATIVE METHODS Part I. Review of Basic Statistics (Chapters 1-11) A. Introduction (Chapter 1) Uncertainty: Decisions are often based on incomplete information from uncertain

### Data Analysis: Describing Data - Descriptive Statistics

WHAT IT IS Return to Table of ontents Descriptive statistics include the numbers, tables, charts, and graphs used to describe, organize, summarize, and present raw data. Descriptive statistics are most

### " Y. Notation and Equations for Regression Lecture 11/4. Notation:

Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through

### Statistical Functions in Excel

Statistical Functions in Excel There are many statistical functions in Excel. Moreover, there are other functions that are not specified as statistical functions that are helpful in some statistical analyses.

### 6 3 The Standard Normal Distribution

290 Chapter 6 The Normal Distribution Figure 6 5 Areas Under a Normal Distribution Curve 34.13% 34.13% 2.28% 13.59% 13.59% 2.28% 3 2 1 + 1 + 2 + 3 About 68% About 95% About 99.7% 6 3 The Distribution Since

### z-scores AND THE NORMAL CURVE MODEL

z-scores AND THE NORMAL CURVE MODEL 1 Understanding z-scores 2 z-scores A z-score is a location on the distribution. A z- score also automatically communicates the raw score s distance from the mean A

### SAMPLING DISTRIBUTIONS

0009T_c07_308-352.qd 06/03/03 20:44 Page 308 7Chapter SAMPLING DISTRIBUTIONS 7.1 Population and Sampling Distributions 7.2 Sampling and Nonsampling Errors 7.3 Mean and Standard Deviation of 7.4 Shape of

### Inferential Statistics

Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

### Business Statistics. Successful completion of Introductory and/or Intermediate Algebra courses is recommended before taking Business Statistics.

Business Course Text Bowerman, Bruce L., Richard T. O'Connell, J. B. Orris, and Dawn C. Porter. Essentials of Business, 2nd edition, McGraw-Hill/Irwin, 2008, ISBN: 978-0-07-331988-9. Required Computing

### Confidence Intervals for Cp

Chapter 296 Confidence Intervals for Cp Introduction This routine calculates the sample size needed to obtain a specified width of a Cp confidence interval at a stated confidence level. Cp is a process

### Chapter 8. Hypothesis Testing

Chapter 8 Hypothesis Testing Hypothesis In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing

### GrowingKnowing.com 2011

GrowingKnowing.com 2011 GrowingKnowing.com 2011 1 Estimates We are often asked to predict the future! When will you complete your team project? When will you make your first million dollars? When will

### Week 3&4: Z tables and the Sampling Distribution of X

Week 3&4: Z tables and the Sampling Distribution of X 2 / 36 The Standard Normal Distribution, or Z Distribution, is the distribution of a random variable, Z N(0, 1 2 ). The distribution of any other normal

### Introduction to Hypothesis Testing. Hypothesis Testing. Step 1: State the Hypotheses

Introduction to Hypothesis Testing 1 Hypothesis Testing A hypothesis test is a statistical procedure that uses sample data to evaluate a hypothesis about a population Hypothesis is stated in terms of the

### Pr(X = x) = f(x) = λe λx

Old Business - variance/std. dev. of binomial distribution - mid-term (day, policies) - class strategies (problems, etc.) - exponential distributions New Business - Central Limit Theorem, standard error

### Introduction to. Hypothesis Testing CHAPTER LEARNING OBJECTIVES. 1 Identify the four steps of hypothesis testing.

Introduction to Hypothesis Testing CHAPTER 8 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 Identify the four steps of hypothesis testing. 2 Define null hypothesis, alternative

### HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION

HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION HOD 2990 10 November 2010 Lecture Background This is a lightning speed summary of introductory statistical methods for senior undergraduate

### Review the following from Chapter 5

Bluman, Chapter 6 1 Review the following from Chapter 5 A surgical procedure has an 85% chance of success and a doctor performs the procedure on 10 patients, find the following: a) The probability that

### Descriptive Statistics

Y520 Robert S Michael Goal: Learn to calculate indicators and construct graphs that summarize and describe a large quantity of values. Using the textbook readings and other resources listed on the web

### Chapter 8 Introduction to Hypothesis Testing

Chapter 8 Student Lecture Notes 8-1 Chapter 8 Introduction to Hypothesis Testing Fall 26 Fundamentals of Business Statistics 1 Chapter Goals After completing this chapter, you should be able to: Formulate

### Introduction to Hypothesis Testing

I. Terms, Concepts. Introduction to Hypothesis Testing A. In general, we do not know the true value of population parameters - they must be estimated. However, we do have hypotheses about what the true

### 10-3 Measures of Central Tendency and Variation

10-3 Measures of Central Tendency and Variation So far, we have discussed some graphical methods of data description. Now, we will investigate how statements of central tendency and variation can be used.

### Chapter 8: Introduction to Hypothesis Testing

Chapter 8: Introduction to Hypothesis Testing We re now at the point where we can discuss the logic of hypothesis testing. This procedure will underlie the statistical analyses that we ll use for the remainder

### Lesson 7 Z-Scores and Probability

Lesson 7 Z-Scores and Probability Outline Introduction Areas Under the Normal Curve Using the Z-table Converting Z-score to area -area less than z/area greater than z/area between two z-values Converting

### Hypothesis testing for µ:

University of California, Los Angeles Department of Statistics Statistics 13 Elements of a hypothesis test: Hypothesis testing Instructor: Nicolas Christou 1. Null hypothesis, H 0 (always =). 2. Alternative

### Point and Interval Estimates

Point and Interval Estimates Suppose we want to estimate a parameter, such as p or µ, based on a finite sample of data. There are two main methods: 1. Point estimate: Summarize the sample by a single number

### Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce

### Hypothesis testing - Steps

Hypothesis testing - Steps Steps to do a two-tailed test of the hypothesis that β 1 0: 1. Set up the hypotheses: H 0 : β 1 = 0 H a : β 1 0. 2. Compute the test statistic: t = b 1 0 Std. error of b 1 =

### Data Analysis. Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) SS Analysis of Experiments - Introduction

Data Analysis Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) Prof. Dr. Dr. h.c. Dieter Rombach Dr. Andreas Jedlitschka SS 2014 Analysis of Experiments - Introduction