# MCQs, Confidence Interval, Probability & Hypothesis Testing

Week 8:

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1. (TCO 9) What is the feasible range for r, the correlation coefficient? (Points: 6)

0 to 1

-0.5 to 0.5

-1 to 1

-1 to 0

2. (TCO 5) The Canadian Services office did a survey of 150 travelers in which they asked if the traveler's first language was French or English. Another question asked was whether the traveler was Canadian born. The results follow.

If a traveler is selected at random (from this group of 150 travelers), find the probability that

P(The traveler is a Canadian.) (Points: 6)

7/15

8/15

13/30

4/17

3. (TCO 5) The Canadian Services office did a survey of 150 travelers in which they asked if the traveler's first language was French or English. Another question asked was whether the traveler was Canadian born. The results follow.

If a traveler is selected at random (from this group of 150 travelers), find the probability that

P(The traveler is not Canadian and speaks French as the first language.) (Points: 6)

20/85

20/70

0.135

20/150

4. (TCO 5) The Canadian Services office did a survey of 150 travelers in which they asked if the traveler's first language was French or English. Another question asked was whether the traveler was Canadian born. The results follow.

If a traveler is selected at random (from this group of 150 travelers), find the probability that

P (The traveler speaks French as a first language given he is a Canadian.) (Points: 6)

65/150

0.800

65/80

65/85

5. (TCO 2) In a study done by the National Health Foundation, it was determined that the mean number of headaches for women is 14 per year with a standard deviation of 2.5.

How many standard deviations is 16.50 from the mean? (Points: 6)

1.25

1.00

0.75

0.50

6. (TCO 6) Find the following probability involving the Standard Normal Distribution. What is P(z < 0.25)? (Points: 6)

0.4013

0.9938

0.8944

0.5987

7. (TCO 8) Suppose you are performing a hypothesis test on a claim about a population proportion. Using an alpha = 0.01 and n = 100, what is the rejection region if the alternate hypothesis is Ha: p > 0.75? (Points: 6)

Reject Ho if z > 1.28

Reject Ho if z > 1.64

Reject Ho if z < -2.33

reject Ho if z > 2.33

8. (TCO 8) Suppose you are performing a hypothesis test on a claim about a population proportion. Using an alpha = 0.05 and n = 90, what two critical values determine the rejection region if the null hypothesis is: Ho: p = 0.35? (Points: 5)

1.28, -1.28

1.96, -1.96

2.33, -2.33

none of these

9. (TCO 2) If the standard deviation of some data is 36, what is the variance? (Points: 5)

6

72

1296

720

10. (TCO 1) Determine the minimum required sample size if you want to be 95% confident that the sample mean is within 3 units of the population mean given sigma = 7.8. Assume the population is normally distributed. (Points: 5)

6

26

13

52

11. (TCO 3) The stem and leaf plot for the following data is displayed below: {70, 78, 76, 55, 43, 56, 32, 67, 68, 71, 75, 67, 60, 62, 58, 75, 21}

Stem and Leaf Plot:

2 | 1

3 | 2

4 | 3

5 | 5 6 8

6 | 0 2 7 7 8

7 | 0 1 5 5 6 8

What is the shape of the data distribution? (Points: 5)

uniform

skewed left

skewed right

symmetric

12. (TCO 4) The random variable X represents the annual salaries in dollars of a group of teachers. Find the expected value E(X).

X = {$30,000, $40,000, $50,000}

P(30,000) = 0.6; P(40,000) = 0.3; P(50,000) = 0.1 (Points: 5)

$18,000

$35,000

$30,000

$22,000

13. (TCO 6) Scores on an exam for entering a private school are normally distributed, with a mean of 72 and a standard deviation of 6. To be eligible to enter, a person must score in the top 5%. What is the lowest score you can earn and still be eligible to enter? (Points: 5)

82

72

80

78

14. (TCO 5) A shipment of 50 television sets contains 3 defective units. How many ways can a vending company buy three of these units and receive no defective units? (Points: 5)

50

10, 210

16,215

12, 324

15. (TCO 6) The time required to process a ton of sugar cane in a plant in Central America in a recent year was normally distributed with a mean 5 days and a standard deviation of 0.5 days (i.e., 12 hours). What is the probability that it will take more than 6 days to process a ton of sugar cane? (Points: 5)

0.9772

0.0228

0.8413

0.1587

16. (TCO 10) The earnings per share (in dollars) for McDonald's Corporation are given by the equation y-hat = 0.668 + 0.046a - 0.015b where 'a' represents total revenue (in billions of dollars) and 'b' represents total net worth (in billions of dollars). Predict the earnings per share when total revenue is $10 billion and net worth is $5 billion. (Points: 5)

1.053

0.778

0.061

0.984

17. (TCO 9) The estimated value for the correlation coefficient for this graph might be

(Points: 3)

-0.50

0.50

1.00

-0.85

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1. (TCO 11) A restaurant claims that its speed of service time is less than 18 minutes. A random selection of 36 service times was collected, and their mean was calculated to be 17.1 minutes. Their standard deviation is 3.1 minutes. Is there enough evidence to support the claim at alpha = 0.08? Perform an appropriate hypothesis test, showing each important step. (Note: 1st Step: Write Ho and Ha; 2nd Step: Determine Rejection Region; etc.) (Points: 20)

2. (TCO 2) The heights of 8 fourth graders are listed in inches: {50, 54, 55, 52, 59, 54, 54, 53}. Find the mean, median, mode, variance, and range. Also, do you think this sample might have come from a normal population? Why or why not? (Points: 20)

3. (TCO 5) A hospital is hoping to introduce a Fast Care Unit. The hospital claims that after the initial run, 78% of the patients were pleased with the service. We ask 10 randomly selected patients whether or not they are pleased with the service.

a) Is this a binomial experiment? Explain how you know.

b) Use the correct formula to find the probability that, out of 10 patients, exactly 7 are pleased with the service. Show your calculations.

(Points: 20)

4. (TCO 6) The average monthly gasoline (gallon) purchase for a family with 2 cars is 62 gallons. This statistic has a normal distribution with a standard deviation = 6 gallons. A family is chosen at random.

a) Find the probability that the family's monthly gasoline (gallon) purchases will be between 47 and 67 gallons.

b) Find the probability that the family's monthly gasoline (gallon) purchases will be less than 67 gallons.

c) What is the probability that the family's monthly gasoline (gallon) purchases will be more than 47 gallons?

(Points: 20)

5. (TCO 7) A marketing firm wants to estimate the average amount spent by patients at the hospital pharmacy. For a sample of 140 randomly selected patients, the mean amount spent was $86.50 and the standard deviation was $11.45.

a. Find a 95% confidence interval for the mean amount spent by patients at the pharmacy. Show your calculations.

b. Interpret this confidence interval and write a sentence that explains it.

(Points: 20)

6. (TCO 7) A drug manufacturer wants to estimate the mean heart rate for patients with a certain heart condition. Because the condition is rare, the manufacturer can only find 19 people with the condition currently untreated. From this small sample, the mean heart rate is 93 beats per minute with a standard deviation of 7.

a. Find a 98% confidence interval for the true mean heart rate of all people with this untreated condition. Show your calculations.

b. Interpret this confidence interval and write a sentence that explains it.

(Points: 20)

7. (TCO 8) For the following statement, write the null hypothesis and the alternative hypothesis. Then, label the one that is the claim being made.

A bus drivers' union claims that the mean age of a bus driver in Chicago is 55.3 years. (Points: 20)

8. (TCO 8) A manufacturer claims that the mean lifetime of its fluorescent bulbs is 1050 hours. A homeowner selects 40 bulbs and finds the mean lifetime to be 1030 hours with a standard deviation of 80 hours. Test the manufacturer's claim. Use alpha = 0.06. (Points: 20)

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#### Solution Summary

The solution provides step by step method for the calculation of testing of hypothesis, confidence interval, normal probability, binomial probability and descriptive statistics. Formula for the calculation and Interpretations of the results are also included. Interactive excel sheet is included. The user can edit the inputs and obtain the complete results for a new set of data.

Point estimate/hypothesis testing...

1. Which of the following is not a characteristic of the normal probability distribution?

a. The mean, median, and the mode are equal

b. The mean of the distribution can be negative, zero, or positive

c. The distribution is symmetrical

d. The standard deviation must be 1

e. None of the above answers is correct.

2. Which of the following is not a characteristic of the normal probability distribution?

a. symmetry

b. The total area under the curve is always equal to 1.

c. 99.72% of the time the random variable assumes a value within plus or minus 1 standard deviation of its mean

d. The mean is equal to the median, which is also equal to the mode.

e. None of the above answers is correct.

3. The level of significance is the

a. Maximum allowable probability of Type II error

b. Maximum allowable probability of Type I error

c. Same as the confidence coefiicient

d. Same as the p-value

e. None of the above answers is correct.

4. If two independent large samples are taken from two populations, the sampling distribution of the difference between the two sample means

a. Can be approximated by a Poisson distribution

b. Will have a variance of one

c. Can be approximated by a normal distribution

d. Will have a mean of one

e. None of the above answers is correct.

5. To compute an interval estimate for the difference between the means of two populations when samples are small, the t distribution can be used if it can be assumed that

a. The populations are normally distributed

b. The variances are equal

c. Both a and b are satisfied

d. The population means are equal

e. None of the above answers is correct.

6. If we are interested in testing whether the mean of population 1 is smaller than the mean of population 2, the

a. Null hypothesis should state µ1 - µ2 < 0

b. Null hypothesis should state µ1 - µ2 > 0

c. Alternative hypothesis should state µ1 - µ2 ≥ 0

d. Alternative hypothesis should state µ1 - µ2 ≤ 0

e. None of the above answers is correct.

7. The pooled variance is appropriate whenever the two populations

a. Are normally distributed

b. Have equal variances

c. Meet both requirements stated in a and b

d. None of the above answers is correct.

8. Ten percent of all employees at a large corporation call in sick on Mondays. A sample of 144 employees' records is taken on a Monday. The probability that the number of employees calling in sick is greater than 22 is

a. 0.0174

b. 0.0244

c. 0.4756

d. 0.9756

e. None of the above answers is correct.

9. The travel time for a college student traveling between her home and her college is uniformly distributed between 40 and 90 minutes. The probability that she will finish her trip in 80 minutes or less is

a. 0.02

b. 0.8

c. 0.2

d. 1

e. None of the above answers is correct.

10. The weight of items produced by a machine is normally distributed with a mean of 8 ounces and a standard deviation of 2 ounces. What is the probability that a randomly selected item will weigh more than 10 ounces?

a. 0.3413

b. 0.8413

c. 0.1587

d. 0.5000

e. None of the above answers is correct.

11. A population has a mean of 180 and a standard deviation of 24. A sample of 64 observations will be taken. The probability that the sample mean will be between 183 and 186 is

a. 0.1359

b. 0.8185

c. 0.3413

d. 0.4772

e. None of these alternatives is correct.

12. From a population of 500 elements, a sample of 225 elements is selected. It is known that the variance of the population is 900. The standard error of the mean is approximately

a. 1.1022

b. 2

c. 30

d. 1.4847

e. 0.45

13. A sample of 92 observations is taken from an infinite population. The sampling distribution of is approximately

a. normal because is always approximately normally distributed

b. normal because the sample size is small in comparison to the population size

c. normal because of the central limit theorem

d. None of these alternatives is correct.

14. A sample of 25 observations is taken from an infinite population. The sampling distribution of p

a. Not normal since n< 30

b. Approximately normal because p is always normally distributed

c. Approximately normal if np ≥ 5 and n(1-P) ≥ 5

d. Approximately normal if np > 30 and n(1-P) > 30

e. None of the above answers is correct.

15. A researcher is interested in determining the average number of years employees of a company stay with the company. If past information shows a standard deviation of 7 months, what size sample should be taken so that at 95% confidence the margin of error will be 2 months or less?

16. The makers of a soft drink want to identify the average age of its consumers. A sample of 16 consumers is taken. The average age in the sample was 22.5 years with a standard deviation of 5 years. Construct an 80% confidence interval for the true average age of the consumers.

17. You are given the following information obtained from a random sample of 4 observations.

25 47 32 56

What is the point estimate of µ?

18. If the standard deviation of the lifetimes of vacuum cleaners is estimated to be 300 hours, how large of a sample must be taken in order to be 97% confident that the margin of error will not exceed 40 hours?

19. Consider the following hypothesis test:

H0: p = 0.5

Ha: p ≠ 0.5

A sample of 800 provided a sample proportion of 0.58.

Using α = 0.05, what is the rejection rule?

20. A new soft drink is being market tested. A sample of 400 individuals participated in the taste test and 80 indicated they like the taste. Determine the p-value.

21. A student believes that the average grade on the statistics final examination is 87. A sample of 36 final examinations is taken. The average grade in the sample is 83.96 with a standard deviation of 12. Compute the probability of a Type II error if the average grade on the final is 85.

22. Identify the null and alternative hypotheses for the following: It has been stated that 75 out of every 100 people who go to the movies on Saturday night buy popcorn.

23. The following information was obtained from independent random samples. Assume normally distributed populations with equal variances.

Sample 1 Sample 2

Sample Mean 45 42

Sample Variance 85 90

Sample Size 10 10

The point of estimate for the standard deviation of the difference between the means of the two populations is

a. 3

b. 4.01

c. 9.37

d. 16.09

e. None of the above answers is correct.

24. The critical F value with 6 numerator and 60 denominator degrees of freedom at a = .05 is

a. 3.74

b. 2.25

c. 2.37

d. None of the above answers is correct.

25. The following information was obtained from matched samples.

Individual Method 1 Method 2

1 7 6

2 5 8

3 6 7

4 7 6

5 5 6

The point estimate for the difference between the means of the two populations is

a. -3

b. -0.6

c. 1

d. 6.3

e. None of the above answers is correct.

26. In order to estimate the difference between the average daily sales of two branches of a department store, the following data has been gathered. Assume the two populations are normally distributed and have equal variances.

Downtown Store North Mall Store

Sample size 12 days 14 days

Sample mean $36,000 $32,000

Sample standard deviation $1,200 $1,000

A point estimate for the difference between the two sample means is

a. 2

b. 200

c. 4000

d. 32000

e. None of the above answers is correct.