# Estimation and Test of Hypothesis

1. When the means of two unrelated samples are used to compare two populations, we are dealing with two dependent means ___ T/F

2. If standard deviation is unknown when completing a hypothesis test about the population mean, then the best estimate for the unknown standard deviation is s. _____ T/F

3. The number of degrees of freedom for the critical value of t is equal to the smaller of n1 -1 or n2 -1 when making inferences about the difference between two independent means for the case when the degrees of freedom are estimated. ___ T/F

4. Pretest versus post-test (before versus after) studies are usually independent samples. ____ T/F

5. The chi-square distribution is used for inferences about the population mean Mean (Mu) when the standard deviation standard deviation is unknown. ____ T/F

6. The t-distribution is used for all inferences about a population's variance. ___ T/F

7. The F distribution is a symmetric distribution. ____ T/F

8. Which of the following would be the correct hypotheses for testing the claim that the mean waiting time to be served at a large post office is at least 6.5 minutes?

A. B. H0: Mean (Mu) = 6.5 and Ha: Mean (Mu) Not Equal To 6.5

B. H0: Mean (Mu) <= 6.5 vs. Ha: Mean (Mu) > 6.5

C. H0: Mean (Mu) >= 6.5 and Ha: Mean (Mu) < 6.5

D. H0: Mean (Mu) > 6.5 vs. Ha: Mean (Mu) >= 6.5

9. Which of the following would be the correct hypotheses for testing the claim that the proportion of male golfers (M) at a particular college is no more than the proportion of female golfers (F) at the college?

A. H0: pM >= pF

B. H0: pM <= pF

C. H0: pM > pF

D. H0: pM < pF

10. The mean age of 25 randomly selected college seniors was found to be 23.5 years, and the standard deviation of all college seniors was 1.3 years. Which of the following is the correct symbol for 23.5 years?

A. Mean (Mu)

B. s (sample)

C. standard deviation (population)

D. X bar

Please show all calculations on every problem.

11. A machine produces 3-inch nails. A sample of 12 nails was selected and the lengths determined. The results are as follows:

2.89 2.95 3.00 3.05 2.99 2.96 3.10 3.06 3.00 3.12 3.00 2.95

Use these results to test H0: Mean (Mu) = 3 and Ha: Mean (Mu) Not Equal To 3 at alpha (a) = 0.05. Give the critical region, the computed test statistic, and the conclusion.

12. A sample of size n = 20 is selected from a normal population to construct a 95% confidence interval estimate for a population mean. The interval was computed to be (8.20 to 9.80). Determine the sample standard deviation.

13. A random sample of 46 observations was selected from a normally distributed population. The sample mean was = 81, and the sample variance was s2 = 35.0. Does the sample show sufficient reason to conclude that the population standard deviation is not equal to 7 at the 0.05 level of significance? Use the p-value method.

14. An insurance company states that 75% of its claims are settled within 5 weeks. A consumer group selected a random sample of 50 of the company's claims and found 35 of the claims were settled within 5 weeks. Is there enough evidence to support the consumer group's claim that fewer than 75% of the claims were settled within 5 weeks? Test using the traditional approach with alpha (a) = 0.05

15. A teacher wishes to compare two different groups of students with respect to their mean time to complete a standardized test. The time required is determined for each group. The data summary is given below. Test the claim at alpha (a) = 0.05, that there is no difference in variance. Give the critical region, test statistic value, and conclusion for the F test.

n1=60, s1=24

n2=120, s2=28

alpha (a) = 0.05

16. A machine produces 9 inch latex gloves. A sample of 80 gloves is selected, and it is found that 20 are shorter than they should be. Find the 99% confidence interval on the proportion of all such gloves that are shorter than 9 inches.

17. The pulse rates below were recorded over a 30-second time period, both before and after a physical fitness regimen. The data is shown below for 10 randomly selected participants. Is there sufficient evidence to conclude that a significant amount of improvement took place? Assume pulse rates are normally distributed. Test using alpha (a) = 0.05.

Before 31 25 30 31 33 25 32 45

After 32 29 30 37 40 35 34 52

18. You are given the following data. Test the claim that there is a difference in the means of the two groups. Use alpha (a) = 0.05.

Group A Group B

x bar 1= 3 x bar 2= 4

s1= 0.5 s2= 0.3

n1= 100 n2= 40

19. Determine the p-value for each of the following hypothesis-testing situations:

a. H0: p = 0.30 and Ha: p Not Equal To 0.30; z test value = 1.65

b. H0: Mean (Mu) >= 30 and Ha: Mean (Mu) < 30; t test value = - 1.7 d.f. = 15

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1 When the means of two unrelated samples are used to compare two populations, we are dealing with two dependent means ___ T/F

F

We are dealing with independent means as the samples are unrelated.

2 If s is unknown when completing a hypothesis test about the population mean, then the best estimate for the unknown standard deviation is s. _____ T/F

T

3 The number of degrees of freedom for the critical value of t is equal to the smaller of n1 -1 or n2 -1 when making inferences about the difference between two independent means for the case when the degrees of freedom are estimated. ___ T/F

F

degrees of freedom = n1+n2-2

4 Pretest versus post-test (before versus after) studies are usually independent samples. ____ T/F

F

These are dependent samples

5 The chi-square distribution is used for inferences about the population mean M when the standard deviation s is unknown. ____ T/F

F

The t distribution is used for inferences about the population mean M when the standard deviation s is unknown and sample size is small.

6 The t-distribution is used for all inferences about a population's variance. ___ T/F

F

The chi square distribution is used for inferences about a population's variance

7 The F distribution is a symmetric distribution. ____ T/F

F

The F distribution is an asymmetric distribution.

8 Which of the following would be the correct hypotheses for testing the claim that the mean waiting time to be served at a large post office is at least 6.5 minutes?

A. B. H0: M = 6.5 and Ha: M not equal to 6.5

B. H0: M <= 6.5 vs. Ha: M > 6.5

C. H0: M >= 6.5 and Ha: M < 6.5

D. H0: M > 6.5 vs. Ha: M >= 6.5

Answer: B. H0: M <= 6.5 vs. Ha: M > 6.5

This is a right tailed test as we are testing the hypothesis that the mean waiting time is more than 6.5 minutes

The rejection region is the right tail.

9 Which of the following would be the correct hypotheses for testing the claim that the proportion of male golfers (M) at a particular college is no more than the proportion of female golfers (F) at the college?

A. H0: pM >= pF

B. H0: pM <= pF

C. H0: pM > pF

D. H0: pM < pF

Answer: A. H0: pM >= pF

The research (alternative) hypothesis is that pM<pF

The null hypothesis H0 is therefore pM>=pF

10 The mean age of 25 randomly selected college seniors was found to be 23.5 years, and the standard deviation of all college seniors was 1.3 years. Which of the following is the correct symbol for 23.5 years?

A. M

B. s

C. s

D.

Answer: D.

23.5 is the

Please show all calculations on every problem.

11 A machine produces 3-inch nails. A sample of 12 nails was selected and the lengths determined. The results are as follows:

2.89 2.95 3 3.05 2.99 2.96 3.1 3.06 3 3.12 3 2.95

Use these results to test H0: M = 3 and Ha: M not equal to 3 at alpha (a) = 0.05. Give the critical region, the computed test statistic, and the conclusion.

We first caculate the sample mean and sample standard deviation

Mean and standard deviation

X= X 2 =

2.89 8.35

2.95 8.70

3.00 9.00

3.05 9.30

2.99 8.94

2.96 8.76

3.10 9.61

3.06 9.36

3.00 9.00

3.12 9.73

3.00 9.00

2.95 8.70

Total= 36.07 108.4693

n=no of observations= 12

Mean= 3.0058 =36.07/12

variance={summation of X 2 - (summation of X) 2 /n}/(n-1)= 0.004445 =(108.4693-36.07^2/12)/(12-1)

standard deviation =square root of Variance= 0.0667 =square root of 0.004445

Data

Hypthesized mean= 3 inches

Sample Standard deviation= 0.0667 inches (calculated above)

Sample mean= 3.0058 inches (calculated above)

Sample size= 12

Significance level= 0.05 0r 5%

Population standard deviation (Known, Not Known)= Not Known

1) Hypothesis

Null Hypothesis: Ho: M = 3 (:Mean is 3 inches)

Alternative Hypothesis: H1: M not equal to 3 :( Mean is not equal to 3 inches )

Significance level=alpha (a) = 0.05 or 5%

No of tails= 2 (Both tails )

This is a 2 tailed (Both tails ) test because we are testing that M not equal to 3

2) Decision rule

sample size=n= 12

Since sample size 12 is less than 30 and poulation standard deviation is 'Not Known' use t distribution

Thus we use t distribution

t at the 0.05 level of significance and 11 degrees of freedom (=n-1) and 2 tailed test= 2.201

t critical = + or - 2.201

if sample statistic is <-2.201 or > 2.201 Reject Null Hypothesis, else Accept Null Hypothesis

Alternatively,

if p value is less than the significance level (= 0.05 ) Reject Null Hypothesis, else Accept Null Hypothesis

3) Calculation of sample statistics

Hypothesized Mean=M = 3 inches

Standard deviation =s= 0.0667 inches

sample size=n= 12

sx=standard error of mean=s/square root of n= 0.0193 = ( 0.0667 /square root of 12)

sample mean= 3.0058 inches

t=(sample mean-M )/sx= 0.3005 =(3.0058-3)/0.0193

4) Compare sample statistic with critical value

test statistic= 0.3005

t critical = + or - 2.201

therefore, sample statistic is within acceptance region

5) Decision

Accept Null Hypothesis (:Mean is 3 inches)

Alternatively, calculate p value

Prob-value corresponding to t = 0.3005 is 76.94% ...

#### Solution Summary

Answers multiple choice / short answer questions on Estimation and Test of Hypothesis.