Chapter 6 Exam A Name The given values are discrete. Use the continuity correction and describe the region of the normal distribution that corresponds to the indicated probability. 1) The probability of exactly 44 green marbles 1) A) The area between 43.5 and 45.5 B) The area between 43.5 and 44 C) The area between 43.5 and 44.5 D) The area between 44 and 44.5 2) A normal quartile plot is given below for a sample of scores on an aptitude test. Use the plot to assess the normality of scores on this test. Explain your reasoning. 2) 3) A study of the amount of time it takes a mechanic to rebuild the transmission for a 2005 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 is less than 8.9 hours. A) 0.9589 B) 0.4276 C) 0.9608 D) 0.9756 3) Round to the nearest tenth unless indicated otherwise. 4) Scores on a test are normally distributed with a mean of 63.2 and a standard deviation of 11.7. Find P81, which separates the bottom 81% from the top 19%. A) 66.6 B) 73.5 C) 0.88 D) 0.291 4) Copyright 2014 Pearson Education, Inc. 1
5) Explain how a nonstandard normal distribution differs from the standard normal distribution. Describe the process for finding probabilities for nonstandard normal distributions. 5) If z is a standard normal variable, find the probability. 6) The probability that z lies between -2.41 and 0 A) 0.5080 B) 0.4910 C) 0.0948 D) 0.4920 6) 7) Complete the following table for a distribution in which μ = 16. It might be helpful to make a diagram to help you determine the continuity factor for each entry. Find the probability that The continuity correction factor is: x is at least 12 x is at most 12 x is more than 12 x is less than 12 7) Find the area of the shaded region. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. 8) 8) -1.82 1.82 z A) 0.4656 B) 0.9656 C) 0.0344 D) -0.0344 9) Scores on a test have a mean of 70 and Q3 is 83. The scores have a distribution that is approximately normal. Find P90. (You will need to first find the standard deviation.) A) 92.9 B) 93.7 C) 95.6 D) 94.8 9) Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. 10) With n = 18 and p = 0.30, estimate P(6). 10) A) 0.1239 B) 0.1015 C) 0.8513 D) 0.1958 11) Assume that z scores are normally distributed with a mean of 0 and a standard deviation of 1. If P(-a < z < a) = 0.4314, find a. A) -0.18 B) 0.3328 C) 0.57 D) 1.49 11) Copyright 2014 Pearson Education, Inc. 2
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. 12) Between 8 pounds and 11 pounds 12) A) 1 2 B) 1 4 C) 2 3 D) 1 3 The given values are discrete. Use the continuity correction and describe the region of the normal distribution that corresponds to the indicated probability. 13) The probability of more than 44 correct answers 13) A) The area to the right of 44.5 B) The area to the right of 43.5 C) The area to the right of 44 D) The area to the left of 44.5 Round to the nearest tenth unless indicated otherwise. 14) A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. Find P60, the score which separates the lower 60% from the top 40%. A) 187.5 B) 212.5 C) 207.8 D) 211.3 14) 15) A normal quartile plot is given below for the weekly incomes (in dollars) of a sample of engineers in one town. Use the plot to assess the normality of the incomes of engineers in this town. Explain your reasoning. 15) 16) Define a density curve and describe the two properties that it must satisfy. Show a density curve for a uniform distribution. Make sure that your graph satisfies both properties. 16) Copyright 2014 Pearson Education, Inc. 3
Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. 17) Estimate the probability of getting exactly 43 boys in 90 births. 17) A) 0.0764 B) 0.0729 C) 0.0159 D) 0.1628 For the binomial distribution with the given values for n and p, state whether or not it is suitable to use the normal distribution as an approximation. 18) n = 24 and p = 0.6 18) A) Normal approximation is suitable. B) Normal approximation is not suitable. 19) For women aged 18-24, systolic blood pressures (in mm Hg) are normally distributed with a mean of 114.8 and a standard deviation of 13.1. If 23 women aged 18-24 are randomly selected, find the probability that their mean systolic blood pressure is between 119 and 122. A) 0.9341 B) 0.3343 C) 0.0577 D) 0.0833 19) 20) The distribution of certain test scores is a nonstandard normal distribution with a mean of 50 and a standard deviation of 6. What are the values of the mean and standard deviation after all test scores have been standardized by converting them to z scores using z = (x - μ)/σ? A) The mean is 100 and the standard deviation is 10. B) The mean is 0 and the standard deviation is 1. C) The mean is 10 and the standard deviation is 100. D) The mean is 1 and the standard deviation is 0. 20) 21) State the central limit theorem. Describe the sampling distribution for a population that is uniform and for a population that is normal. 21) Use the normal distribution to approximate the desired probability. 22) Find the probability that in 200 tosses of a fair die, we will obtain at most 30 fives. A) 0.1871 B) 0.2946 C) 0.4936 D) 0.3229 22) 23) The number of books sold over the course of the four-day book fair were 194, 197, 247, and 76. Assume that samples of size 2 are randomly selected with replacement from this population of four values. List the different possible samples, and find the mean of each of them. 23) 24) Lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. (a) Find the probability of a pregnancy lasting more than 250 days. (b) Find the probability of a pregnancy lasting more than 280 days. Draw the diagram for each and discuss the part of the solution that would be different to finding the requested probabilities. 24) Copyright 2014 Pearson Education, Inc. 4
Using the following uniform density curve, answer the question. 25) What is the probability that the random variable has a value greater than 5? A) 0.325 B) 0.250 C) 0.375 D) 0.500 25) Copyright 2014 Pearson Education, Inc. 5
Answer Key Testname: CHAPTER 6 EXAM A 1) C 2) Since the normal quartile plot is roughly linear, it appears that scores on this test are approximately normally distributed. 3) C 4) B 5) Both distributions are bell-shaped. In a standard distribution, the mean is 0 and the standard deviation is 1. In a nonstandard distribution, the mean and standard deviation can have other values. (Of course, SD values are positive.) The process should include sketching the diagram, marking the x score on the diagram, computing the corresponding z score, shading the appropriate area of interest, finding the area corresponding to the z score in the table (the area between the mean and the z-score), then using that area to find the shaded area of interest. 6) D 7) Find the probability that The continuity correction factor is: x is at least 12 11.5 x is at most 12 12.5 x is more than 12 12.5 x is less than 12 11.5 8) B 9) D 10) D 11) C 12) A 13) A 14) B 15) Since the normal quartile plot displays curvature, it appears that incomes of engineers in this town are probably not normally distributed. 16) A density curve is a graph of a continuous probability distribution satisfying these two properties: 1) The total area under the curve must be 1. 2) Every point on the curve must have a vertical height that is 0 or greater. Answers to the graphs may vary, but the total area under the curve must be 1. 17) A 18) A 19) C 20) B 21) Assume that the random variable x has a distribution (which may or may not be normal) with mean μ and standard deviation σ. Samples of size n are randomly selected from this population. 1) The distribution of sample means x will, as the sample size increases, approach a normal distribution. 2) The mean of the sample means will be the population mean μ. σ 3) The standard deviation of the sample means will be n. For both the uniform and the normal distribution, the distribution of the sample means is bell-shaped. As n gets larger, the distribution approaches the normal distribution. In the case where the original population is uniform, the distribution of the sample means is approximately normal for n > 30. 22) B 23) Possible samples: 194-194; 194-197; 194-247; 194-76; 197-194; 197-197; 197-247; 197-76; 247-194; 247-197; 247-247; 247-76; 76-194; 76-197; 76-247; 76-76 Means: 194, 195.5, 220.5, 135, 195.5, 197, 222, 136.5, 220.5, 222, 247, 161.5, 135, 136.5, 161.5, 76 24) Both (a) and (b) are solved by subtracting the area score from 1. 25) C Copyright 2014 Pearson Education, Inc. 6