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MECH 320 Practice Problems Not to be graded

1. When 1094 American households were surveyed, it was found that 67% of them owned two cars.

Identify the population and the sample.

2. A recent survey by the alumni of a major university indicated that the average salary of 10,500 of its

175,000 graduates was $95,000. Determine whether the numerical values is a parameter or a statistic.

3. Identify whether following statements describe inferential or descriptive statistics.

a. The average age of the students in a statistics class is 19 years.

b. There is a relationship between smoking cigarettes and getting emphysema.

4. The top 14 speeds, in miles per hour, for Pro-Stock drag racing over the past two decades are listed

below. Find the mean speed and sample standard deviation.

181.1 202.2 190.1 201.4 191.3 201.4 192.2 201.2 193.2 201.2 194.5 199.2 196.0 196.2

5. What is the difference between using μ and ̅ to represent a mean?

6. You need to purchase a battery for your car. There are two types available. Type A has a mean life of

five years and a standard deviation of one year. Type B has a mean life of five years and a standard

deviation of one month. Both batteries cost the same. Which one should you purchase if you are

concerned that your car will always start? Explain your reasoning.

7. Heights of adult women have a mean of 63.6 in. and a standard deviation of 2.5 in. Does Chebyshevʹs

Theorem say about the percentage of women with heights between 58.6 in. and 68.6 in.?

8. Find the normalized Z variable for a normal distributed X=55 with a mean of 58 and the standard

deviation of 3.

9. Test scores for a history class had a mean of 79 with a standard deviation of 4.5. Test scores for a

physics class had a mean of 69 with a standard deviation of 3.7. Suppose a student gets an 83 on the

history test and a 84 on the physics test. On which test did the student perform better?

10. Classify the events as dependent or independent. Events A and B where

a. P(A) = 0.7, P(B) = 0.7, and P(A and B) = 0.49

b. P(A) = 0.8, P(B) = 0.1, and P(A and B) = 0.07

11. From the probability distribution, find the mean and standard deviation for the random variable x,

which represents the number of cars per household in a town of 1000 households.

x 0 1 2 3 4

P(x) 0.125 0.428 0.256 0.108 0.083


PART II. For the following questions, identify the type of the probability distribution (discrete vs.

continuous, and name) and answer the following questions:

1. A company is interested in evaluating its current inspection procedure on shipments of 50

identical items. The procedure is to take a sample of 5 and pass the shipment if no more than 2 are

found to be defective. What proportion of 20% defective shipments will be accepted?

2. Find the probability that a person flipping a coin gets the third head on the seventh flip.

3. A manufacturing company uses an acceptance scheme on production items before they are

shipped. The plan is a two-stage one. Boxes of 25 are readied for shipment and a sample of 3 are

tested for defectives. If any defectives are found, the entire box is sent back for 100% screening.

If no defectives are found, the box is shipped.

a. What is the probability that a box containing three defectives will be shipped?

b. What is the probability that a box containing only one defective will be sent back for

screening?

4. On the average a certain intersection results in three traffic accidents per week. What is the

probability that exactly five accidents will occur at this intersection in any given week?

5. The probability that a person dies from a certain respiratory infection is 0.002. Find the

probability that fewer than 5 of the next 2000 so infected will die.

6. From a lot of 10 missiles, 4 are selected at random and fired. If the lot contains 3 defective

missiles that will not fire, what is the probability that at most 2 will not fire?

7. Suppose that airplane engines operate independently in flight and fail with probability q = 1/5.

Assuming that a plane makes a safe flight if at least half of its engines run, determine whether a

four-engine plane or a two-engine plane has the highest probability for a successful flight.

8. A scientist inoculates several mice, one at a time, with a disease germ until he finds two that have

contracted the disease. If the probability of contracting the disease is 1/6, what is the probability

that eight mice are required?

9. In testing a certain kind of truck tire over a rugged terrain, it is found that 25% of the trucks

fail to complete the test run without a blowout. What is the probability that from 5 to 10 of

the next 15 trucks tested have flat tires? How many of the 15 trucks tested would you expect to

have a flat tire? Using Chebychev’s theorem, find and interpret the interval .

Partial Solutions

1) X » Hypergeometric (M = 10;N = 50; n = 5); P(X · 2)

2) X » Negative Binomial (r = 3; p = .5); P(X = 4)

3) a) X » Hypergeometric (M = 3;N = 25; n = 3); P(X = 0)

b) X » Hypergeometric (M = 1;N = 25; n = 3); P(X = 1)

4) X » Poisson ( = 3); P(X = 5)

5) X » Binomial (n = 2000; p = .002) approx: X » Poisson ( = 4); 6) X » Hypergeometric (M = 3;N = 10; n = 4); a) P(X = 0) b) 7) X » Binomial (n = 4; p = .80); and X » Binomial (n = 2; p = .80); 8) X » Negative Binomial (r = 2; p = 1/6 ); P(X = 6)

9) X » Binomial (n = .15; p = .25);

p = .25; E(X) = 1:25; = 0:9375; √


PART III:

1. Find the area of the indicated region under the standard normal curve.

2. Find the probability of z occurring in the indicated region.

3. The lengths of pregnancies of humans are normally distributed with a mean of 268 days and a standard

deviation of 15 days. Find the probability of a pregnancy lasting more than 300 days.

4. A mathematics professor gives two different tests to two sections of his college algebra courses. The

first class has a mean of 56 with a standard deviation of 9 while the second class has a mean of 75 with a

standard deviation of 15. A student from the first class scores a 62 on the test while a student from the

second class scores an 83 on the test. Compare the scores.


5. In a certain normal distribution, find the standard deviation σ when μ = 50 and 10.56% of the area lies

to the right of 55.

6. The amounts of time employees of a telecommunications company have worked for the company are

normally distributed with a mean of 5.1 years and a standard deviation of 2.0 years. Random samples of

size 18 are drawn from the population and the mean of each sample is determined.

7. Decide if it is appropriate to use the normal distribution to approximate the random variable x for a

binomial experiment with sample size of n = 7 and probability of success p = 0.2.

8. For a sample of 20 IQ scores the mean score is 105.8. The standard deviation, σ, is 15. Determine

whether a normal distribution or a t-distribution should be used or whether neither of these can be used to

construct a confidence interval. Assume that IQ scores are normally distributed.

A) Use normal distribution.

B) Use the t-distribution.

C) Cannot use normal distribution or t-distribution.

9. In a random sample of 28 families, the average weekly food expense was $95.60 with a standard

deviation of $22.50. Determine whether a normal distribution or a t-distribution should be used or

whether neither of these can be used to construct a confidence interval. Assume the distribution of weekly

food expenses is normally shaped.

A) Use the t-distribution.

B) Use normal distribution.

C) Cannot use normal distribution or t-distribution.

10. Let X be a random variable with mean and variance . What is the mean, , and variance,

, of

the random variable defined by ? (a and b are constant parameters).

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