# Probability, Regression Analysis & Hypothesis Testing

1. (A) Classify the following as an example of nominal, ordinal, interval, or ratio level of measurement, and state why it represents this level: zip codes for the state of Pennsylvania

(B) Determine if this data is qualitative or quantitative: Nationality

(C) In your own line of work, give one example of a discrete and one example of a continuous random variable, and describe why each is continuous or discrete

2. A newsmagazine asks 1200 students what college they attend, and how many times per week they attend a party where alcohol is served. The news magazine determines the mean number of parties per week for each school, and publishes a ranking of the nation's top ten party schools.

I. What is the population?

II. What is the sample?

III. Is the study observational or experimental? Justify your answer.

IV. What are the variables?

V. For each of those variables, what level of measurement (nominal, ordinal, interval, or ratio) was used to obtain data from these variables?

3. 3. Construct both an ungrouped and a grouped frequency distribution for the data given below:

171 169 168 174 180 172 172 170 168 178

169 166 174 171 169 170 177 173 172 175

4. Given the following frequency distribution, find the mean, variance, and standard deviation. Please show all of your work.

Class Frequency

56-58 25

59-61 21

62-64 8

65-67 13

68-70 21

5. The following data lists the average monthly snowfall for January in 15 cities around the US:

8 35 31 26 36 41 29 40

17 16 33 38 30 34 13

Find the mean, variance, and standard deviation. Please show all of your work.

6. Rank the following data in increasing order and find the positions and values of both the 32nd percentile and 85th

percentile. Please show all of your work.

0 5 3 0 5 2 1 5 7 3 5 2

7. For the table that follows, answer the following questions:

x y

1 1/4

2 1/2

3 3/4

4

- Would the correlation between x and y in the table above be positive or negative?

- Find the missing value of y in the table.

- How would the values of this table be interpreted in terms of linear regression?

- If a "line of best fit" is placed among these points plotted on a coordinate system, would the slope of this line be positive or negative?

8. Determine whether each of the distributions given below represents a probability distribution. Justify your answer.

(A)

x 1 2 3 4

P(x) 11/25 6/25 3/25 1/5

(B)

x 3 6 8

P(x) 0.4 1/5 5/10

(C)

x 20 30 40 50

P(x) 0.41 0.07 0.32 0.2

9. A set of 50 data values has a mean of 15 and a variance of 36.

I. Find the standard score (z) for a data value = 30.

II. Find the probability of a data value > 30.

III. Find the probability of a data value < 30.

Show all work

10. Answer the following:

(A) Find the binomial probability P(x = 5), where n = 14 and p = 0.50.

(B) Set up, without solving, the binomial probability P(x is at most 5) using probability notation.

(C) How would you find the normal approximation to the binomial probability P(x = 5) in part A? Please show how you would calculate µ and σ in the formula for the normal approximation to the binomial, and show the final formula you would use without going through all the calculations

11. Assume that the population of heights of male college students is approximately normally distributed with mean  of 72.15 inches and standard deviation  of 6.39 inches. A random sample of 96 heights is obtained. Show all work.

(A) Find P(x > 73.25) _

(B) Find the mean and standard error of the X distribution _

(C) Find P(X > 73.25)

(D) Why is the formula required to solve (A) different than (C)?

12. Determine the critical region and critical values for z that would be used to test the null hypothesis at the given level of significance, as described in each of the following:

(A) Ho:μ = 86 and Ha:μ ≠86  = 0.10

(B) Ho:μ ≤79 and Ha:μ <79  = 0.05

(C) Ho:μ ≥70 and Ha:μ >70  = 0.01

13. Describe what a type I and type II error would be for each of the following null hypotheses:

Ho: There is no good plan for the Iraq war

14. A researcher claims that the average age of people who buy theatre tickets is 49. A sample of 30 is selected and their ages are recorded as shown below. The standard deviation is 7. At  = 0.05 is there enough evidence to reject the researcher's claim? Show all work.

50 46 54 48 52 49 46 44 48 53

44 49 57 60 58 51 56 50 55 53

45 52 45 60 52 59 54 59 51 59

15. Write a correct null and alternative hypothesis for testing the claim that the mean life of a battery for a cell phone is at least 75 hours.

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#### Solution Summary

The solution provides step by step method for the calculation of binomial, normal probabilities, regression analysis and hypothesis Testing. Formula for the calculation and Interpretations of the results are also included.

Final Questions: 13 Statistics problems

1. Routine physical examinations are conducted annually as part of a health service program for the employees. It was discovered that 8% of the employees needed corrective shoes, 15% needed major dental work and 3% needed both corrective shoes and major dental work. What is the probability that an employee selected at random will need either corrective shoes or major dental work?

A) 0.20

B) O.25

C)0.50

D)1.00

E) None of the above

2. A lamp manufacturer has developed five lamp bases and four lampshades that could be used together. How many different arrangements of base and shade can be offered?

A) 5

B) 10

C) 15

D) 20

Chapter 7

3. The mean amount spent by a family of four on food per month is $500 with a standard deviation of $75. Assuming that the food costs are normally distributed, what is the probability that a family spends less than $410 per month?

A) 0.2158

B) 0.8750

C) 0.0362

D) 0.1151

Chapter 9

4. A state meat inspector in Texas has been given the assignment of estimating the mean and net weight of packages of ground turkey labeled "3 pounds." Of course, she realizes that the weights cannot be precisely 3 pounds. She takes a sample of 36 packages, and found that the mean weight is 3.01 pounds, with a standard deviation of 0.03 pounds. Determine a 95% confidence interval for the population mean.

Chapter 10

5) Texas Agriculture Real Estate Company specializes in selling farm property in the state of Texas. Its records show that the mean selling time of farm property is 90 days. Because of recent drought conditions, the agency believes that the mean selling time is now greater than 90 days. A statewide survey of 100 farms sold recently revealed that the mean selling time was 94 days with a sample standard deviation of 22 days. At the .01 significance level and using the steps of hypothesis testing, can the company conclude there has been an increase in selling time?

a) One tailed or Two tailed? _________________

b) Step 1: State the null hypothesis and the alternate hypothesis.

Null: There has not been an increase in the selling time of 90 days.

Alternate: There has been an increase in the selling time of 90 days.

H0 : μ ______ 90 (what symbol goes in each)

H1 : μ ______ 90

c) Step 2: Select the level of significance. ______________

d) Step 3: Select the test statistic. t or z? Why? ________________________________________________________

e) Step 4: Formulate a decision rule. What is the cut off? ________________

f) Calculate the z or t:

g) Step 5: Make a decision and (h) interpret the result. (what can you conclude?)

i) What is the p-value?

6) A recent survey of college freshman revealed that the average freshman got 7 hours of sleep per night. A random sample of 50 students at TAMU-Commerce showed the mean number of hours slept the night before was 6 hours and 48 (6.8 hours), with a standard deviation of the sample of .9 hours. At the .05 significance level and using the steps of hypothesis testing, can TAMU-C conclude that their freshman sleep less than the typical American freshman?

a) One tailed or Two tailed? _________________

b) Step 1: State the null hypothesis and the alternate hypothesis.

Null: Students get 7 or more hours of sleep.

Alternate: Students get less than 7 hours of sleep.

H0 : μ ______ 7 (what symbol goes in each)

H1 : μ ______ 7

c) Step 2: Select the level of significance. ______________

d) Step 3: Select the test statistic.

t or z? Why? ________________________________________________________

e) Step 4: Formulate a decision rule.

What is the cut off? ________________

f) Calculate the z or t:

g) Step 5: Make a decision and (h) interpret the result. (what can you conclude?)

i) What is the p-value?

Chapter 11

7) Maxwell House Coffee is interested in whether the mean daily consumption of regular-coffee drinkers is less than that of decaf-coffee drinkers. Assume the population standard deviation for those drinking regular coffee is 1.20 cups per day and 1.36 cups per day for those drinking decaf. A random sample of 50 regular-coffee drinkers showed a mean of 4.35 cups per day. A sample of 40 decaf drinkers showed a mean of 5.84 cups per day. Using the steps in hypothesis testing, is there a significant difference in the two populations of coffee drinkers? (.05 significance level)

a) One tailed or Two tailed? _________________

b) Step 1: State the null hypothesis and the alternate hypothesis.

Population 1 (regular coffee drinkers) = μ1

Population 2 (decaf coffee drinkers) = μ2

Null: There is no difference between the mean number of cups for regular coffee drinkers and decaf drinkers.

Alternate: The two means of the two types of drinkers is not equal.

H0 μ1 ______ μ2 (what symbol goes in each)

H1 μ1 ______ μ2

c) Step 2: Select the level of significance. ______________

d) Step 3: Select the test statistic.

t or z? Why? ________________________________________________________

e) Step 4: Formulate a decision rule.

What is the cut off? ________________

f) Calculate the z or t:

g) Step 5: Make a decision and (h) interpret the result. (what can you conclude?) From a marketing standpoint, what could Maxwell House conclude about the coffee drinkers and how should they market their coffee?

i) What is the p-value?

8) Cingular offers two plans to its subscribers: Plan A and Plan B. When new subscribers sign up, they are asked to provide demographic information. From this information, Cingular has found the following: The mean yearly income for a sample of 40 subscribers to Plan A is $57,000 with a sd of $9,200; The mean yearly income for a sample of 30 subscribers to Plan B is $61,000 with a sd of $7,100. At the .05 significance level and using the steps to hypothesis testing, is it reasonable to conclude the mean income of those selecting Plan B is larger than Plan A?

a) One tailed or Two tailed? _________________

b) Step 1: State the null hypothesis and the alternate hypothesis.

Population 1 (Plan A) = μA

Population 2 (Plan B) = μB

Null: Plan B is not larger than Plan A.

Alternate: Plan B is larger than Plan A.

H0 μA ______ μB (what symbol goes in each)

H1 μA ______ μB

c) Step 2: Select the level of significance. ______________

t or z? Why? ________________________________________________________

e) Step 4: Formulate a decision rule.

What is the cut off? ________________

f) Calculate the z or t:

g) Step 5: Make a decision and (h) interpret the result. (what can you conclude?)

i) What is the p-value?

Chapter 13

9. Assume the least squares equation is Y' = 10 + 20X. What does the value of 10 in the equation indicate?

A) Y intercept

B) For each unit increased in Y, X increases by 10

C) For each unit increased in X, Y increases by 10

D) None of the above

Use the following to answer questions:

A sales manager for an advertising agency believes there is a relationship between the number of contacts and the amount of the sales. To verify this believe, the following data was collected:

Sales Person Number of Contacts Sales (in thousands)

1 14 24

2 12 14

3 20 28

4 16 30

5 46 80

6 23 30

7 48 90

8 50 85

9 55 120

10 50 110

10. What is the dependent variable?

A) Salesperson

B) Number of contacts

C) Amount of sales

D) All the above

11. What is the independent variable?

A) Salesperson

B) Number of contacts

C) Amount of sales

D) All the above

12. What is the Y-intercept of the linear equation?

A) -12.201

B) 2.1946

C) -2.1946

D)12.201

13. What is the slope of the linear equation?

A) -12.201

B) 12.201

C) 2.1946

D) -2.1946

14. What is the value of the coefficient of correlation?

A) 0.6317

B) 0.9754

C) 0.9513

D) 9.3104

See attached file.

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