Basic Statistics 42.202,

Fall2003

Exam II

October 30th, 2003

Solution

Question 1:[64 Points] For each question in this section, circle the correct answer.

  1. In a large population of adults, the mean IQ is 112 with a standard deviation of 20. Suppose 200 adults are randomly selected for a market research campaign. The distribution of the sample mean IQ is

(a)Exactly normal with mean 112 and standard deviation 20.

(b)Approximately normal with mean 112 and standard deviation of 0.1.

(c)Approximately normal with mean 112 and standard deviation of 1.414.

(d)Approximately normal with mean 112 and standard deviation of 20.

  1. The weights of extra large eggs have a normal distribution with a mean of 1 oz. and a standard deviation of 0.1 oz. The probability that a dozen eggs weighs more than 13 oz. is closest to

(a) / 0.0000 / (b) / 0.0020 / (c) / 0.1814 / (d) / 0.2033

Questions 3-5 refer to the following information:

The student body of a large university consists of 60% female students. A random sample of 8 students is selected.

3.What is the probability that among the students in the sample exactly two are female?

(a) / 0.0896 / (b) / 0.2936 / (c) / 0.0413 / (d) / 0.0007

4.What is the probability that among the students in the sample at least 7 are female?

(a) / 0.1064 / (b) / 0.0896 / (c) / 0.0168 / (d) / 0.8936

5.The mean and the variance of the female students in the sample are

(a) / 4.8 and 1.92 / (b) / 3.2 and 1.92 / (c) / 4.8 and 1.39 / (d) / 3.2 and 1.39

6.The central limit theorem states if the sample size n is large then

(a)The distribution of the observations (the sample data) is approximately normal.

(b)The distribution of the sample mean is approximately standard normal.

(c)The distribution of the sample mean is approximately normal.

(d)The distribution of the population mean is approximately normal.

7.An airplane is only allowed a gross passenger weight of 8000 kg. If the weights of passengers traveling by air between Toronto and Vancouver have a mean of 78 kg and a standard deviation of 7 kg, the approximate probability that the combined weight of 100 passengers will exceed 8,000 kg is:

(a) / 0.4978 / (b) / 0.3987 / (c) / 0.0044 / (d) / 0.0022

8. An experiment consists of four outcomes with P(E1)=0.20, P(E2)=0.30 and P(E3)=0.40, then P(E4)=

(a) / 0.50 / (b) / .024 / (c) / 0.10 / (d) / 0.90

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9. All human blood can be typed as one of O, A, B, or AB, but the distribution of types varies a bit with race. Here is the probability model for the blood type of a randomly chosen black American:

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Blood type / O / A / B / AB
Probability / 0.49 / 0.27 / 0.20 / ?

The probability that a randomly chosen black American has blood type O or type AB is

(a) / 0.69 / (b) / 0.59 / (c) / 0.49 / (d) / 0.53

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10. The scores of the 12th-grade students on the national Assessment of Educational Progress year 2000 mathematics test have a distribution that is approximately normal with and standard deviation. Now choose SRS of 36 12th-graders. The probability that their mean score higher than 310 is

(a) / 0.3906 / (b) / 0.0478 / (c) / 0.9522 / (d) / 0.6094

11.The average hourly wage at a fast food restaurant is $5.85 with a standard deviation of $0.35. Assume that the wages are normally distributed. The probability that the average hourly wage of a sample of 25 workers is greater than $5.92 is

(a) / 0.5793 / (b) / 0.4207 / (c) / 0.1587 / (d) / 0.8413

12.Government regulations indicate that the total weight of cargo in a certain kind of airplane cannot exceed 330 kg. On a particular day a plane is loaded with 100 boxes of goods. If the weight distribution for individual boxes is normal with mean 3.2 kg and standard deviation 0.4 kg, what is the probability that the regulations will not be met:

(a) / 0.4938 / (b) / 0.1239 / (c) / 0.0062 / (d) / 0.9938

13.The length of time patients must wait to see a doctor in a local clinic is uniformly distributed between 15 minutes and 2.5 hours. The probability that a patient would have to wait between 45 minutes and 2 hours is

(a) / 0.5556 / (b) / 0.2222 / (c) / 0.5000 / (d) / 0.2907

Questions 14-16 refer to the following information:

Here are the proportions of earned degrees in the United States in a recent year, classified by level and by the sex of degree recipient:

Bachelor’s / Master’s / Professional / Doctorate
Female / 0.3788 / 0.1193 / 0.0185 / 0.0098
Male / 0.3253 / 0.1052 / 0.0271 / 0.0160

14.If you select a degree recipient at random, the probability that the selected person is a woman is

(a) / 0.8560 / (b) / 0.5264 / (c) / 0.7196 / (d) / 0.3788

15.The conditional probability that you choose a woman, given that the person chosen received a professional degree is

(a) / 0.4054 / (b) / 0.5264 / (c) / 0.0350 / (d) / 0.0185

16.The events “choose a woman” and “choose a professional degree recipient” are

(a)Independent.

(b)Disjoint.

(c)Not independent.

(d)Independent and disjoint.

Section II:[36 Points] Free-Response Problems

Question (1): [4+4+4+4=16 points]

Question (2): [4+4+4+4+4=20 points]

MNM Corporation gives each of its employees an aptitude test. The scores on the test are normally distributed with a mean of 75 and a standard deviation of 15. A simple random sample of 25 is taken from a population of 500.

a.What are the expected value, the standard deviation, and the shape of the sampling distribution of?

(73,3), Normal

b.What is the probability that the average aptitude test in the sample will be between 70.14 and 82.14?

0.9387

c.What is the probability that the average aptitude test in the sample will be greater than 82.68?

0.0052

d.What is the probability that the average aptitude test in the sample will be less than 78.69?

0.8907

e.Find a value, C, such that P( C) = .015.

81.51

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