Chapter 6 - Practice Problems 1

Transcription

1 Chapter 6 - Practice Problems 1 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Using the following uniform density curve, answer the question. 1) What is the probability that the random variable has a value greater than 6? 1) A) B) C) D) ) What is the probability that the random variable has a value less than 4? 2) A) B) C) D) ) What is the probability that the random variable has a value between 2 and 7.6? 3) A) B) C) D) Assume that the weight loss for the first month of a diet program varies between 6 pounds and 12 pounds, and is spread evenly over the range of possibilities, so that there is a uniform distribution. Find the probability of the given range of pounds lost. 4) More than 10 pounds 4) A) 5 6 B) 1 7 C) 2 3 D) 1 3 5) Between 7 pounds and 10 pounds 5) A) 1 2 B) 1 4 C) 2 3 D) 1 3 If Z is a standard normal variable, find the probability. 6) The probability that Z lies between 0 and ) A) B) C) D) ) The probability that Z is less than ) A) B) C) D) ) The probability that Z lies between and ) A) B) C) D) ) P(Z < 0.97) A) B) C) D) ) 1

2 10) P(-0.73 < Z < 2.27) A) B) C) 1.54 D) ) The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0 C at the freezing point of water. Tests on a large sample of these thermometers reveal that at the freezing point of water, some give readings below 0 C (denoted by negative numbers) and some give readings above 0 C (denoted by positive numbers). Assume that the mean reading is 0 C and the standard deviation of the readings is 1.00 C. Also assume that the frequency distribution of errors closely resembles the normal distribution. A thermometer is randomly selected and tested. Find the temperature reading corresponding to the given information. 11) Find P96, the 96th percentile. 11) A) 1.03 B) C) 1.82 D) ) Find Q3, the third quartile. 12) A) 0.67 B) -1.3 C) 0.82 D) ) If 7% of the thermometers are rejected because they have readings that are too high, but all other thermometers are acceptable, find the temperature that separates the rejected thermometers from the others. 13) A) 1.39 B) 1.48 C) 1.45 D) ) A quality control analyst wants to examine thermometers that give readings in the bottom 4%. Find the reading that separates the bottom 4% from the others. 14) A) B) C) D) ) For a standard normal distribution, find the percentage of data that are more than 1 standard deviation away from the mean. 15) A) 68.26% B) 15.87% C) 31.74% D) 34.13% 16) For a standard normal distribution, find the percentage of data that are between 3 standard deviations below the mean and 1 standard deviation above the mean. 16) A) 16.00% B) 15.74% C) 99.74% D) 84.00% 17) For a standard normal distribution, find the percentage of data that are more than 2 standard deviations below the mean or more than 3 standard deviations above the mean. 17) A) 2.41% B) 97.59% C) 0.26% D) 4.56% 18) If a continuous uniform distribution has parameters of μ = 0 and σ = 1, then the minimum value for the co ntinuous variable is - 3 and the maximum value is 3. For this distribution, find P(-1.5 < x < 1). Round your answer to three decimal places. 18) A) B) C) D)

3 Assume that X has a normal distribution, and find the indicated probability. 19) The mean is μ = 60.0 and the standard deviation is σ = 4.0. Find the probability that X is less than ) A) B) C) D) ) The mean is μ = 15.2 and the standard deviation is σ = 0.9. Find the probability that X is between 14.3 and ) A) B) C) D) ) In one region, the September energy consumption levels for single-family homes are found to be normally distributed with a mean of 1050 kwh and a standard deviation of 218 kwh. Find P45, which is the consumption level separating the bottom 45% from the top 55%. 21) A) B) C) D) ) The amount of rainfall in January in a certain city is normally distributed with a mean of 4.4 inches and a standard deviation of 0.6 inches. Find the value of the quartile Q1. A) 4.2 B) 1.1 C) 4.0 D) ) Find the indicated probability. 23) The diameters of bolts produced by a certain machine are normally distributed with a mean of 0.30 inches and a standard deviation of 0.01 inches. What percentage of bolts will have a diameter greater than 0.32 inches? 23) A) 97.72% B) 47.72% C) 2.28% D) 37.45% 24) The volumes of soda in quart soda bottles are normally distributed with a mean of 32.3 oz and a standard deviation of 1.2 oz. What is the probability that the volume of soda in a randomly selected bottle will be less than 32 oz? 24) A) B) C) D) ) Scores on a test have a mean of 66 and Q3 is 81. The scores have a distribution that is approximately normal. Find the standard deviation. Round your answer to the nearest tenth. 25) A) 11.3 B) 20 C) 22.4 D) ) A math teacher gives two different tests to measure studentsʹ aptitude for math. Scores on the first test are normally distributed with a mean of 24 and a standard deviation of 5.7. Scores on the second test are normally distributed with a mean of 72 and a standard deviation of Assume that the two tests use different scales to measure the same aptitude. If a student scores 28 on the first test, what would be his equivalent score on the second test? (That is, find the score that would put him in the same percentile.) 26) A) 76 B) 80 C) 79 D) 77 3

4 27) A history teacher assigns letter grades on a test according to the following scheme: 27) A: Top 10% B: Scores below the top 10% and above the bottom 60% C: Scores below the top 40% and above the bottom 20% D: Scores below the top 80% and above the bottom 10% F: Bottom 10% Scores on the test are normally distributed with a mean of 69 and a standard deviation of Find the numerical limits for each letter grade. A) A: Above 80 B: Between 72 and 80 C: Between 59 and 71 D: Between 53 and 58 F: Below 53 C) A: Above 80 B: Between 72 and 80 C: Between 59 and 71 D: Between 55 and 58 F: Below 55 B) A: Above 85 B: Between 72 and 85 C: Between 59 and 71 D: Between 53 and 58 F: Below 53 D) A: Above 85 B: Between 75 and 85 C: Between 59 and 74 D: Between 53 and 58 F: Below 53 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 28) A recent survey based on a random sample of n = 410 voters, predicted that the Independent candidate for the mayoral election will get 25% of the vote, but he actually gets 28%. Can it be concluded that the survey was done incorrectly? 28) 29) A poll of 1500 randomly selected students in grades 6 through 8 was conducted and found that 36% enjoy playing sports. Is the 36% result a statistic or a parameter? Explain. 29) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Identify the probability of each sample, and describe the sampling distribution of the sample means. 30) Personal phone calls received in the last three days by a new employee were 4, 2, and 7. Assume that samples of size 2 are randomly selected with replacement from this population of three values. 30) A) 1/9; 2/9; 1/9; 0/9; 1/9; 0/9; 1/9; 2/9; 1/9 B) 1/8; 1/9; 1/8; 1/9; 1/8; 1/9; 1/8; 1/9; 1/8 C) 1/9; 1/9; 1/9; 1/9; 1/9; 1/9; 1/9; 1/9; 1/9 D) 1/3; 1/3; 1/18; 1/6; 1/18; 1/9; 1/6; 1/18; 1/9 31) The amount of snowfall falling in a certain mountain range is normally distributed with a mean of 86 inches, and a standard deviation of 12 inches. What is the probability that the mean annual snowfall during 36 randomly picked years will exceed 88.8 inches? 31) A) B) C) D)

5 32) The weights of the fish in a certain lake are normally distributed with a mean of 15 lb and a standard deviation of 9. If 9 fish are randomly selected, what is the probability that the mean weight will be between 12.6 and 18.6 lb? 32) A) B) C) D) ) Assume that womenʹs heights are normally distributed with a mean of 63.6 inches and a standard deviation of 2.5 inches. If 90 women are randomly selected, find the probability that they have a mean height between 62.9 inches and 64.0 inches. 33) A) B) C) D) ) A study of the amount of time it takes a mechanic to rebuild the transmission for a 1992 Chevrolet Cavalier shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. If 40 mechanics are randomly selected, find the probability that their mean rebuild time exceeds 8.7 hours. 34) A) B) C) D) ) A study of the amount of time it takes a mechanic to rebuild the transmission for a 1992 Chevrolet Cavalier shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. If 40 mechanics are randomly selected, find the probability that their mean rebuild time exceeds 7.7 hours. 35) A) B) C) D) ) In a population of 225 women, the heights of the women are normally distributed with a mean of 64.5 inches and a standard deviation of 2.9 inches. If 25 women are selected at random, find the probability that their mean height will exceed 66 inches. Assume that the sampling is done without replacement and use a finite population correction factor with N = ) A) B) C) D) ) In a population of 150 women, the heights of the women are normally distributed with a mean of 64.3 inches and a standard deviation of 3.2 inches. If 36 women are selected at random, find the mean μ x and standard deviation σ x of the population of sample means. Assume that the 37) sampling is done without replacement and use a finite population correction factor. A) 56.2 inches, 2.8 inches B) 64.3 inches, 3.2 inches C) 64.3 inches, 1.88 inches D) 64.3 inches, 0.47 inches SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 38) A population consists of these values: 3, 5, 8, 14, 17. Find the mean, μ, and standard deviation, σ, of the population. List all samples of size n = 2 that are obtained without replacement. Find the population of all values of x by finding the mean of each sample that you listed. Find the mean μ x and standard deviation σ x for the population of sample 38) means by using the formulas: σ μ x = μ and σ x = n N - n N - 1 5

6 39) When sampling without replacement from a finite population of size N, the following formula is used to find the standard deviation of the population of sample means: σ x = σ n N - n N - 1 However, when the sample size n, is smaller than 5% of the population size, N, the finite N - n population correction factor,, can be omitted. Explain in your own words why N - 1 this is reasonable. For N = 200, find the values of the finite population correction factor when the sample size is 10%, 5%, 3%, 1% of the population, respectively. What do you notice? 39) 6

7 Answer Key Testname: CH 6 SET 1 1) C 2) B 3) D 4) D 5) A 6) C 7) B 8) B 9) A 10) D 11) D 12) A 13) B 14) C 15) C 16) D 17) A 18) A 19) A 20) A 21) B 22) C 23) C 24) C 25) C 26) C 27) B 28) No, because of sampling variability, sample proportions will naturally vary from the true population proportion, even if sampling is done with a perfectly valid procedure. 29) Statistic, because it is calculated from a sample, not a population. 30) C 31) D 32) C 33) D 34) A 35) C 36) A 37) D 38) μ = 9.4, σ = 5.314; possible samples are: (3, 5), (3, 8), (3, 14), (3, 17), (5, 8), (5, 14), (5, 17), (8, 14), (8, 17), (14, 17); population of values of x : 4, 5.5, 8.5, 10, 6.5, 9.5, 11, 11, 12.5, 15.5; μ x = 9.4 and σ x = ) Answers will vary. Possible answer: when the sample is a very small portion of the population, the selection of each item, even though it is not replaced, will have very little effect on the nature of the population. In this case, sampling without replacement will be approximately equivalent to sampling with replacement. Values of the finite population correction factor when the sample size is 10%, 5%, 3%, 1% of the population are 0.951, 0.977, 0.987, respectively. As the sample size drops as a percentage of the population, the finite population correction factor approaches 1, which confirms that it is reasonable to omit it for small sample sizes 7