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# Statistics problems

#3 - Use your own paper as an answer sheet. Show all mathematical calculations. No partial credit will be given if work is not shown. A calculator will be required to complete the exam.

1. Find the 90% confidence interval for a population mean if the sample mean is

21.6 with a standard deviation of 2.8 for a sample of 35.

2. Find the 95% confidence interval for a population mean if the sample mean is 146 with a standard deviation of 12 for a sample of 18.

3. Find the minimum sample size to estimate a population mean if the population

standard deviation is 5.6, the confidence interval is 99%, and the maximum error

of estimate is 4.

4. Find a 90% confidence interval for the population proportion if a sample of 106

had a sample proportion of 21%.

5. Find the minimum sample size to find a 95% interval estimate for a population

proportion if previous sample had a sample proportion of 35% and you want to be

accurate within 3% of the true proportion.

6. Find a 90% confidence interval for the population variance if a sample of 12 had a variance of 360.

7. A car gets a mean of 18 mpg. Test to see if a cheaper grade of gasoline made any difference in the gas miles at the .01 level. A sample of 38 fillups with the

cheaper gas got a mean of 17.8 mpg and a standard deviation of 0.6. State the

hypotheses and identify the claim, find the critical value(s), compute the test

value, make the decision, summarize the results.

8. An emergency team responds to calls in a mean time of 8.4 minutes with a

standard deviation of .7. A new dispatching system is being tested to see if the

response time can be lowered. If a sample of 18 calls with the new system were

answered in a mean time of 7.8 minutes, test at the .05 level. State the hypotheses

and identify the claim, find the critical value(s), compute the test value, make the

decision, summarize the results.

9. A motorist claims that the South Boro Police issue an average of 60 speeding

tickets per day. The following data show the number of speeding tickets each day

for a period of one month. Assume the population standard deviation is 13.42. Is

there enough evidence to reject the motorist`s claim at a significance level of .05?

Use the P value approach.

State the hypotheses and identify the claim, find the critical value(s), compute the

test value, make the decision, summarize the results. A calculator/computer

software may be used to find the sample mean and standard deviation.

72 45 36 68 69 71 57 60

83 26 60 72 58 87 48 59

60 56 64 68 42 57 57 58

63 49 73 75 42 63

10. A certain lottery claims that 10% of its tickets win some prize. A sample of 50

tickets had 3 winners. At the .01 level, test the claim that the proportion is 10%.

State the hypotheses and identify the claim, find the critical value(s), compute the

test value, make the decision, summarize the results.

11. Test at the .01 level the claim that the population variance is less than 158 if a

sample of 10 had a variance of 164. State the hypotheses and identify the claim,

find the critical value(s), compute the test value, make the decision, summarize

the results.

12. A ski-shop manager claims that the average of the sales for her shop is $1800 a day during the winter months. Ten winter days are selected at random, and the

mean of the sales is $1830. The standard deviation of the population is $200.

Can one reject the claim at a significance level of .05? Find the 95% confidence

interval of the mean. Does the confidence interval interpretation agree with the

hypothesis test results? Explain. Assume that the variable is normally distributed?

13. Brand X and Brand Y of pain relievers were tested to see how much ibuprofen

each tablet contained. Does Brand X have more ibuprofen? Test at the .05 level.

BRAND X BRAND Y

n1 = 36 n2 = 35

1 x = 358 2 x = 345

s1 = 10 s2 = 14

14. A study of teenagers found the following information on the length of time(in

minutes) each talked on the telephone. Find the 95% confidence interval of the

true differences in means.

Boys Girls

n1 = 50 n2 = 50

1 x = 358 2 x = 345

s1 = 10 s2 = 14

15. A consumer advocate claims that there is no difference in the variance of the

number of hours that two companies` batteries will last. A sample of 10 batteries

is selected from company X, and the variance of hours is 24. A sample of 10

batteries from company Y has a variance of 40. At a significance level of 0.10,

test the claim that there is no difference in the variance of the life of the batteries.

16. A bank studied the mean time clerks spent with a customer. A sample of 13 male clerks spent a mean time of 1.4 minutes with each customer with s = .31. A

sample of 14 female clerks spent a mean time of 1.7 minutes with each customer

with s = .16. Be sure to test for equality of variances first. Test at the .05 level

the claim that men and women clerks spend the same amount of time with each

customer.

17. A study was conducted of pregnant women who smoked and those who did not smoke, to see if maternal smoking had an effect on the birth weight. At the .05

level, test the claim that maternal smoking can lower the birth weight of a child.

Assume the variances to be equal.

Smokers Non-smokers

n1 = 10 n 2 = 10

1 x = 6.9 2 x = 7.2

s1 = .5 s 2 = .4

18. Students were paired by matching their IQs and grades in previous mathematics courses taken. Group A attended lecture and did homework. Group B watched videotapes and did work on a computer. Test to see if there is a significant

difference in the final exam scores for the two groups at the .05 level.

Lecture Tapes

95 99

87 91

91 88

85 90

81 87

79 78

74 79

73 81

71 65

69 74

19. In a class of remedial English that was taught by the lecture method, 52 out of 75 students completed the course with a passing grade. In the same course with

computer-assisted instruction, 69 out of 95 completed the course with a passing

grade. At the .05 level, is there a significant difference in the passing rates for the

two different methods of instruction?

20. Use the following data to:

a) draw a scatter plot

b) find the coefficient of correlation and test the significance at the .05 level

c) find the regression line

d) predict y′ for x = 5

e) find the coefficients of determination and non-determination

f) find the standard error of estimate

g) find the 95% confidence interval for the number of alcoholic drinks of

x = 5

Number of alcoholic drinks - x Score on a Dexterity Test - y

2 15

1 18

3 11

4 7

2 10

1 16

5 4

6 2

21. A real estate agent found that there is a significant relationship among the number of acres on a farm ( x 1), the number of rooms in the farmhouses ( x 2 ) , and the

selling price in ten thousands ( y ) of farms in a specific area. The regression

equation is y′ = 44.9 - 0.0266 x 1 + 7.56 x 2. Predict the selling price of a farm

that has 500 acres and a farmhouse with 8 rooms.

#4 -

1. An instant oatmeal mix is considering adding flavors to its mix. 200 people tested the flavors and gave their preferences. Is there a preference for the flavor at the

.05 level? State the hypotheses and identify the claim, find the critical value(s),

compute the test value, make the decision, summarize the results.

Plain 20

Cinnamon 58

Apple 48

Maple 22

Peach 52

2. A USA Today poll shows that 74% of respondents felt that other motorists were driving more aggressively than they did five years ago, 23% felt that other

motorists were driving the same way they did five years ago, and 3% felt other

motorists were driving less aggressively than they were driving five years ago. A

sample survey of 180 senior drivers found that 125 felt that other motorists were

driving more aggressively than they did five years ago, 36 felt that other motorists

were driving about the same as they did five years ago, and 19 felt that other

motorists were driving less aggressively than they did five years ago. At a

significance level of .10 test the claim that senior drivers feel the same way as

those who were surveyed in the USA Today poll. State the hypotheses and

identify the claim, find the critical value(s), compute the test value, make the

decision, summarize the results.

3. Given the following information, is the grade dependent on gender? Test at the

.05 level. State the hypotheses and identify the claim, find the critical value(s),

compute the test value, make the decision, summarize the results.

A B C D or lower

Males 8 17 25 10

Females 12 10 21 7

4. Suppose a researcher wants to investigate whether the proportion of smokers

within different age groups is the same. He divides the American population into

four age groups. Within each age group, he surveys 80 individuals and asks,

"Have you smoked at least one cigarette in the past week?". Test at the .01 level.

State the hypotheses and identify the claim, find the critical value(s), compute the

test value, make the decision, summarize the results. The results of the survey are

as follows:

Age

18-29 30-49 50-64 65 or older

Smoked at least one cigarette in past week 24 21 23 12

Did not smoke at least one cigarette in past week 56 59 57 68

Exercises 5

If the null hypothesis is rejected in exercise 5 use the Scheffe` test if the sample sizes are unequal to test the differences between the means, and use the Tukey test if the sample sizes are equal. For this exercise, state the hypotheses and identify the claim, find the critical value(s), compute the test value, make the decision, summarize the results.

5. The weights in ounces of four types of women`s shoes are shown here. At

α = 0.05 test the claim that there is no difference in the mean weights of the

groups.

Dress Heals Dress Flats Casual Heels Casual flats

8 6 11 6

7 6 12 9

7 7 12 7

6

1 x = 7 2 x = 6.3 3 x = 11.7 4 x = 7.3

s1 = .82 s2 = .58 s3 = .58 s4 = 1.53

6. A record store employee claims that the median age of the customers is 17. Test

at the .01 level if the population was not normally distributed. State the

hypotheses and identify the claim, find the critical value(s), compute the test

value, make the decision, summarize the results.

The data are shown here:

15 16 14 13 12 11 18 19

20 21 25 27 16 15 14 12

7. A group of dieters started an exercise program. Is there a difference in the weight

loss without and with exercise at .05 level. Use the paired sample sign test. State

the hypotheses and identify the claim, find the critical value(s), compute the test

value, make the decision, summarize the results.

Average weight loss per week

Dieter No exercise With exercise

1 2 3

2 3 3.5

3 1 1.5

4 2.5 3

5 4 6

6 2 4

7 4 3

8 5 6.5

9 5 5

10 4 6.5

8. A group of men and women smokers were asked at what age they started

smoking. Using the Wilcoxon rank sum test at the .01 level, determine if there is

a significant difference in the starting ages, based on gender. State the hypotheses

and identify the claim, find the critical value(s), compute the test value, make the

decision, and summarize the results.

Females Males

14 14

15 12

22 15

18 16

17 16

22 13

19 12

20 18

21 17

13 19

9. Students were asked to rate a teacher on a scale of 1 to 20. The students were

grouped by the grade they made in the class. Is there a difference in the ranks

given by students with different grades? Test at the .01 level. Assume that the

population is not normally distributed, so use the Kruskal-Wallis test. State the

hypotheses and identify the claim, find the critical value(s), compute the test

value, make the decision, summarize the results.

A or B C D or F

16 19 12

17 18 14

18 16 15

15 14 10

16 19 12

17

10. Children and adults ranked six different soft drinks. Us the Spearman rank

correlation coefficient to test if there is a relationship between how adults and

children rank soft drinks. Test at the .05 level. State the hypotheses and identify

the claim, find the critical value(s), compute the test value, make the decision, and

summarize the

results.

Brand Children Adults

A 1 2

B 2 1

C 5 5

D 4 4

E 3 3

F 6 6

11. Are these answers for a true-false test in random order? State the hypotheses and identify the claim, find the critical value(s), compute the test value, make the

decision, summarize the results.

T T F T F F T T T F F T F F T T F F T T

Use this chart that shows a number of hours worked per week and the number of credit hours enrolled in for 50 students with part time jobs.

Student # Hours Worked Credit Hour Student # Hours Worked Credit Hours

1. 20 14 26 38 9

2. 28 13 27. 35 12

3. 13 16 28. 10 18

4. 20 16 29. 12 21

5. 20 18 30. 10 18

6. 20 15 31. 20 12

7. 15 6 32. 20 17

8. 38 13 33. 20 12

9. 24 13 34. 40 12

10. 20 17 35. 20 18

11. 22 13 36. 28 13

12. 12 9 37. 12 9

13. 28 13 38. 24 18

14. 3 12 39. 24 18

15. 8 6 40. 6 18

16. 38 13 41. 10 17

17. 20 18 42. 40 13

18. 20 15 43. 4 13

19. 20 21 44. 18 16

20. 38 13 45. 25 13

21. 36 12 46. 20 17

22. 10 9 47. 32 17

23. 28 13 48 12 18

24. 25 16 49. 18 12

25. 10 13 50. 15 15

12. Use random sampling to pick 10 students and find the mean number of hours

worked.

13. Use systematic sampling to find the mean hours worked for a sample of 10

students.

14. Use the hours worked to divide the students into 4 groups according to the

number of hours worked (1-10 hours, 22-20 hours, 21-30 hours, 31-40 hours).

Decide how many to take from each group to take a stratified sample of 10 and

find the mean hours worked.