One-Tailed Test Procedure (σ Known)

Transcription

1 One-Tailed Test Procedure (σ Known) State the hypotheses: Draw a picture H 0 : μ μ 0 ; H a : μ < μ 0 OR H 0 : μ μ 0 ; H a : μ > μ 0 Compute the test statistic: z = x μ 0 σ/ n Compute the p-value (the area to the left of z, if lower tailed, the area to the right of z, if upper tailed) Decide whether or not to reject H 0. (Reject if p-value α OR Reject if z z α, for upper-tailed if z z α, for upper-tailed )

2 Two-Tailed Tests (σ Known)

3 Two-Tailed Test (σ Known) Hypotheses: H 0 : μ = μ 0 ; H a : μ μ 0 Test statistic: z = x μ 0 σ/ n Note: the rejection area is now split (half and half) on both tails of the curve So the p-value is twice the area. Reject H 0 if: z z α/2 or z z α/2 z α/2 0 z α/2

4 Two-Tailed Test Procedure (σ Known) State the hypotheses: H 0 : μ = μ 0 ; H a : μ μ 0 Draw a picture Compute the test statistic: z = x μ 0 σ/ n Compute the p-value (twice the area to the left of z, if z is negative; twice the area to the right of z, if z is positive) Decide whether or not to reject H 0. (Reject if p-value α OR Reject if z z α/2 )

5 So Duh! example, company perspective H 0 : μ = 20 H a : μ 20 20

6 So Duh! problem, company perspective Assume, still, that the population standard deviation is known to be 0.5 ounces. We need to adjust our processes if it appears that we are not filling the bottles correctly. I.E. if we get a sample mean far away from the expected mean of 20 oz. We don t want to go through the expense of fully auditing all of our processes if they re actually performing correctly, but we re to accept a 10% chance of that happening. (That s our level of significance possibility of a Type I error.)

7 So Duh! problem, company perspective Assume we create a random sample 50 bottles of soda and find that the sample mean volume is ounces. Do we audit our processes? μ x = μ = 20 σ x = σ n = Compute the corresponding z for z = =

8 So Duh! problem, company perspective Assume we create a random sample 50 bottles of soda and find that the sample mean volume is ounces. Do we audit our processes? μ x = μ = 20 σ x = σ n = Compute the corresponding z for 19.8 z = = The area to the right of z is

9 So Duh! problem, company perspective The area to the right of z ( ) is There was only a = % chance of getting a sample with a mean that far away from 20 if 20 were actually our population mean

10 So Duh! problem, company perspective Since 8.92% < 10% we reject H 0 Equivalently, since we reject H 0

11 Try This The mean Math + Verbal SAT score for Winthrop students in Fall 2013 was We randomly sample 50 Winthrop students to determine if the mean SAT is different than it was in a. State the null and alternative hypotheses. b. For our sample, the mean Math + Verbal SAT score is Assuming a population standard deviation of 112 points, find the p- value. c. Using a 5% significance level, what is your conclusion?

12 Hypothesis Tests: Population Mean, when σ is Unknown If we do not know the population standard deviation, σ, then we must use the sample standard deviation, s, as an estimate for it. In this case we will us a t-distribution (instead of a z-distribution) with n 1 degrees of freedom Assumes the population is normally distributed, but this method is still valid for most populations if n 30, and even for highly skewed populations if n 50 t = x μ 0 s/ n

13 Hypothesis Test for μ Procedure (σ Unknown) State the hypotheses: H 0 : μ μ 0 ; H a : μ < μ 0 OR H 0 : μ μ 0 ; H a : μ > μ 0 OR H 0 : μ = μ 0 ; H a : μ μ 0 Draw a picture Compute the test statistic: t = x μ 0 s/ n Compute/Estimate the p-value Lower: area to left of t Upper: area to right of t Two-Tailed: t > 0, twice area to right t < 0, twice area to left Decide whether or not to reject H 0. Reject if p-value α

14 A 2012 report from the Bureau of Labor Statistics reported that U.S. workers had an average job tenure of 4.6 years. A random sample of 54 U.S. workers is surveyed to see if the average tenure has increased. a. State the null and alternative hypotheses. H 0 : μ 4.6; H a : μ > 4.6 b. If the sample mean is 4.72 years with a sample standard deviation of 0.75 years, compute the p-value. t = / the area to the right of t, the p-value, is between 0.1 and 0.2 a. Using a 10% significance level, what should be the researcher s conclusion? since the p value is bigger than 0.1, do not reject H 0

15 A local fast food restaurant found last year that the mean drive through wait time was 3.64 minutes. The staff has changed quite a bit since last year, so a random sample of 40 drive through wait times is compiled to determine if the mean wait time has changed. a. State the null and alternative hypotheses. b. If the sample mean is 3.39 minutes with a sample standard deviation of 1.18 minutes, compute the p-value. a. Using a 10% significance level, what should be the researcher s conclusion?

16 Hypothesis Tests: Population Proportion Recall the sampling distribution of the sample proportion is normally distributed about the population proportion can be approximated by a normal distribution if np 5 and n(1 p) 5 with: μ p = p σ p = p(1 p) n

17 Hypothesis Tests: Population Proportion So, we have the following three types of hypothesis tests Lower Tail H 0 : p p 0 ; H a : p < p 0 Upper Tail H 0 : p p 0 ; H a : p > p 0 Two-Tailed H 0 : p = p 0 ; H a : p p 0 For each, we ll use σ p = p 0(1 p 0 ) n Test statistic is z = p p 0 p0(1 p0) n

18 Hypothesis Test for Population Proportion State the hypotheses: H 0 : p p 0 ; H a : p < p 0 OR H 0 : p p 0 ; H a : p > p 0 OR H 0 : p = p 0 ; H a : p p 0 Draw a picture Compute the test statistic:z = p p 0 p0(1 p0) n Compute/Estimate the p-value Lower: area to left of z Upper: area to right of z Two-Tailed: z > 0, double area to right z < 0, double the area to left Decide whether or not to reject H 0. Reject if p-value α

19 A 2014 Pew Research poll found that 70% of Facebook uses stated that they visit the site every day. A random sample of 150 Facebook users is surveyed to see if that percentage has changed. a. State the null and alternative hypotheses. H 0 : p = 0.7; H a : p 0.7 b. If 97 of the Facebook users in our sample say that they visit the site every day, compute the p-value. z = (1 0.7) the area to the left of z is , so our p-value is a. Using a 10% significance level, what should be the researcher s conclusion? since the p value is bigger than 0.1, do not reject H 0

20 Only 1.5% of free application casual gamers made an in-app purchase in January A sample of 500 of these casual gamers is surveyed to see if the percentage who made in-app purchases in a month has increased. a. State the null and alternative hypotheses. b. If 11 of the gamers in our sample made and in-app purchase in August, find the p-value. a. Using a 10% significance level, what should be the researcher s conclusion?