# For a randomly selected student in a particular high school, let Y denote the number of living grandparents of the student.

For a randomly selected student in a particular high school, let Y denote the number of living grandparents of the student. What are the possible values of the random variable Y? (Points: 15)

1,2,3,4

4

0,1,2,3,4

0,1,2

2. When two balanced dice are rolled, 36 equally likely outcomes are possible as shown:

1,1 1,2 1,3 1,4 1,5 1,6

2,1 2,2 2,3 2,4 2,5 2,6

3,1 3,2 3,3 3,4 3,5 3,6

4,1 4,2 4,3 4,4 4,5 4,6

5,1 5,2 5,3 5,4 5,5 5,6

6,1 6,2 6,3 6,4 5,6 6,6

Let X denote the smaller of the two numbers. If both dice come up the same number, the X equals that common value. Find the probability distribution of X. Leave your probabilities in fractional form.

(Points: 15)

x P(X=x)

1 5/18

2 2/9

3 1/6

4 1/9

5 1/18

6 0

x P(X=x)

1 11/36

2 1/4

3 7/36

4 5/36

5 1/12

6 1/36

x P(X=x)

1 1/6

2 1/6

3 1/6

4 1/6

5 1/6

6 1/6

x P(X=x)

1 5/18

2 1/4

3 7/36

4 5/36

5 1/9

6 1/36

3. Find the mean of the random variable.

The random variable X is the number of people who have a college degree in a randomly selcted group of four adults from a particular town. Its probability distribution is given in the table:

x P(X=x)

0 0.4096

1 0.4096

2 0.1536

3 0.0256

4 0.0016

(Points: 15)

2.00

1.21

0.80

0.70

4. Find the standard deviation of the random variable.

The random variable X is the number of people who have college degrees in a randomly selected group of four adults from a particular town. Its probability distribution is given in the table:

x P(X=x)

0 0.2401

1 0.4116

2 0.2646

3 0.0756

4 0.0081

(Points: 15)

1.51

1.06

0.84

0.92

5. Evaluate the expression:

(Points: 15)

720

210

2880

3

6. Find the probability of at least 2 girls in 6 births. Assume that the male and female births are equally likely and that the births are independent events. (Points: 15)

0.891

0.656

0.234

0.109

7. On a multiple choice test with 14 questions, each question has four possible answers, one of which is correct. For students who guess at all the answers, find the mean for the random bvariable X, the number of correct answers. (Points: 15)

10.5

3.5

4.7

7

8. Find the mean of the binomial random variable

The probability that a radish seed will germinate is 0.7. A gardener plants seeds in batches of 11. Find the mean for the random variable X, the number of seeds germinating in each batch.

(Points: 15)

7.81

9.9

7.7

3.3

9. Find the confidence interval specified. Assume that the opoulation is normally distributed.

A laboratory tested twelve chicken eggs and found that the mean amount of cholesterol was 230 milligrams with s = 14.3 milligrams. Construct a 95% confidence interval for the tru mean choleserol content of all such eggs.

(Points: 15)

220.9 to 239.1 milligrams

221.0 to 239.0 milligrams

220.8 to 239.2 milligrams

222.6 to 237.4 milligrams

10. For samples of the specified size from the population described, find the mean and standard deviation of the sample mean .

The National Weather Service keeps records of snowfall in mountain ranges. Records indicate that in a certain range, the annual snowfall has a mean of 86 inches and a standard deviation of 14 inches. Suppose the snowfalls are sampled during randomly picked years. For samples of size 49, determine the mean and standard deviation of .

(Points: 15)

= 86; = 2

= 14; = 86

= 2; = 86

= 86; = 14

11. Find the requested value.

A researcher wishes to estimate the mean resting heart rate for long-distance runners. a random sample of 12 long-distance runners yeilds the following heart rates, in beats per minute:

80 81 61 79 59 66

80 62 72 78 60 63

Use the data to obtain a point estimate of the mean resting heart rate for all long-distance runners.

(Points: 15)

71.9 beats per minute

68.4 beats per minute

70.1 beats per minute

66.7 beats per minute

12. Identify the distribution of the sample mean. In particular, state whether the distribution of is normal or approximately normal and give its mean and standard deviation.

Let x represent the number which shows up when a balanced die is rolled. Then x is a random variable with a mean of 3.5 and a standard deviation of 1.71. Let denote the mean of the numbers obtained when the die is rolled 34 times. Determine the sampling distribution of .

(Points: 15)

Normal, mean = 3.5, standard deviation = 0.29

Normal, mean = 3.5, standard deviation = 0.05

Approximately normal, mean = 3.5, standard deviation = 1.71

Approximately normal, mean = 3.4, standard deviation = 0.29

13. Find the value of α that corresponds to a level of confidence of 93% (Points: 15)

0.93

0.07

0.007

7

14. Classify the hypothesis as two-tailed, left-tailed, or right -tailed

In the past, the mean running time for a certain type of flashlight battrery has been 8.3 hours. the manufacturer has introduced a change in the production method and wants to perform a hypothesis test to determine whether the mean running time has changed as a result.

(Points: 15)

Right-tailed

Two-tailed

Left-tailed

15. Fill in the blanks by standardizing the normally distributed variable.

Dave drives to work each morning at about the same time. His commute time is normally distributed with a mean of 42 minutes and a standard deviation of 4 minutes. The percentage of time that his commute lies between 50 and 54 minutes is equal to the area under the standard normal curve between _____________ and _____________.

(Points: 15)

1.5 - 2.5

0 - 1

2.5 - 3.5

2 - 3

16. Use a table of areas to find the specified area under the standard normal curve.

The area that lies between 0 and 3.01

(Points: 15)

0.9987

0.4987

0.1217

0.5013

17. Find the confidence interval specified.

The mean score , on an aptitude test for a random sample of 6 students was 65. Assuming that σ =12, construct a 95.44% confidence interval for the mean score, μ , of all the students taking the test.

(Points: 15)

57.7 to 72.3

41 to 89

61.0 to 74.8

55.2 to 74.8

18. Determine the margin of error in estimating the population mean, μ.

A sample of 45 eggs yeilds a mean weight of 1.37 ounces. Assuming that σ = 0.53 ounces, find the margin of error in estimating μ at the 95% level of confidence.

(Points: 15)

0.13 oz.

0.02 oz.

0.15 oz.

6.71

19. The graph portrays the decision criteria for a hypothesis test for a population mean. The null hypothesis is . The curve is the normal curve for the test statistic under the assumption that the null hypothesis is true. Use the graph to solve the problem

A graphical display of the decision criteris follows:

Determine the rejection region.

(Points: 15)

All z-scores lie to the right of 2.575

All z-scores lie to the left of 2.575

0.005

2.575

20. Use a table of areas for the standard normal curve to find the required z-score.

Find the z-score for which the area under the standard normal curve to its left is 0.40

(Points: 10)

.25

-.25

-.57

.57

21. Find the indicated probability or percentage for the normally distributed variable.

The variable X is normally distributed. The mean is μ =22.0 and the standard deviation is σ = 2.4. find P(19.7 < X < 25.3)

(Points: 10)

0.3370

1.0847

0.7477

0.4107

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