Question

A) Suppose a random sample of size 22 is taken from a normally distributed population, and the sample mean and variance are calculated to be x¯=5.26 and s2=0.5respectively.

Use this information to test the null hypothesis H0:μ=5 versus the alternative hypothesis HA:μ>5.

a) What is the value of the test statistic t, for testing the null hypothesis that the population mean is equal to 5? Round your response to at least 3 decimal places.

b) The P-value falls within which one of the following ranges:

A		P-value > 0.10
B		0.05 < P-value < 0.10
C		0.025 < P-value < 0.05
D		0.01 < P-value < 0.025
E		P-value < 0.01

c) Is the null hypothesis rejected at the 5% level of significance? Yes or No

d) Is the null hypothesis rejected at the 1% level of significance? Yes or No

B) Suppose a random sample of size 15 is taken from a normally distributed population, and the sample mean and variance are calculated to be x¯=5.14 and s2=6.5respectively.

Use this information to test the null hypothesis H0:μ=5versus the alternative hypothesis HA:μ<5, at the 10% level of significance.

a) What is the value of the test statistic, for testing the null hypothesis that the population mean is equal to 5? Round your response to at least 3 decimal places.

b)The p-value falls within which one of the following ranges:

A		p-value > 0.10
B		0.05 < p-value < 0.10
C		0.025 < p-value < 0.05
D		0.01 < p-value < 0.025
E		p-value < 0.01

c) Is the null hypothesis rejected at the 10% level of significance? Yes or No

C) Which of the following statements are true?

Note that there may be more than one correct answer; select all that are true.

1		The χ2 statistic can only take on values greater than or equal to 0.
2		Because the χ2 distribution is asymmetric, we must always conduct two-sided hypothesis tests.
3		A χ2 value can be used to find a two sided P-value by considering twice the amount of area under the curve in the corresponding tail. The P-value can be at most 1.
4		When running a hypothesis test for standard deviation, if the sample size is large enough, our sample can come from a population with a non-Normal distribution

D) A random sample of size 12 is taken from a normally distributed population, and a sample variance of 16.0 is calculated.

Use this information to test the null hypothesis H0:σ2=15 against the alternative hypothesis HA:σ2≠15.

a)What is the value of the test statistic χ2? Round your response to at least 3 decimal places.

b)The P-value falls within which one of the following ranges:

A		P-value > 0.10
B		0.05 < P-value < 0.10
C		0.025 < P-value < 0.05
D		0.01 < P-value < 0.025
E		P-value < 0.01

c) What conclusion can be made, at the 5% level of significance?

A		There is insufficient evidence to reject the null hypothesis, and therefore no significant evidence that the population variance is not 15.
B		There is sufficient evidence to reject the null hypothesis, in favour of the alternative hypothesis that the population variance is not 15.

Answer 1

A) Suppose a random sample of size 22 is taken from a normally distributed population, and the sample mean and variance are calculated to be x¯=5.26 and s2=0.5respectively.Use this information to test the null hypothesis H0:μ=5 versus the alternative hypothesis HA:μ>5.

t Test for Hypothesis of the Mean
Data
Null Hypothesis                m=	5
Level of Significance	0.05
Sample Size	22
Sample Mean	5.26
Sample Standard Deviation	0.707106781
Intermediate Calculations
Standard Error of the Mean	0.1508
Degrees of Freedom	21
t Test Statistic	1.7246
Upper-Tail Test
Upper Critical Value	1.7207
p-Value	0.0496
Reject the null hypothesis

a) What is the value of the test statistic t, for testing the null hypothesis that the population mean is equal to 5? Round your response to at least 3 decimal places.

t=1.725

b) The P-value falls within which one of the following ranges:

A            P-value > 0.10

B 0.05 < P-value < 0.10

Answer: C 0.025 < P-value < 0.05

D 0.01 < P-value < 0.025

E P-value < 0.01

c) Is the null hypothesis rejected at the 5% level of significance? Yes

d) Is the null hypothesis rejected at the 1% level of significance? No

B) Suppose a random sample of size 15 is taken from a normally distributed population, and the sample mean and variance are calculated to be x¯=5.14 and s2=6.5respectively.

Use this information to test the null hypothesis H0:μ=5versus the alternative hypothesis HA:μ<5, at the 10% level of significance.

t Test for Hypothesis of the Mean
Data
Null Hypothesis                m=	5
Level of Significance	0.1
Sample Size	15
Sample Mean	5.14
Sample Standard Deviation	2.549509757
Intermediate Calculations
Standard Error of the Mean	0.6583
Degrees of Freedom	14
t Test Statistic	0.2127
Lower-Tail Test
Lower Critical Value	-1.3450
p-Value	0.5827
Do not reject the null hypothesis

1. What is the value of the test statistic, for testing the null hypothesis that the population mean is equal to 5? Round your response to at least 3 decimal places.

t = 0.213

b)The p-value falls within which one of the following ranges:

Answer: A                         p-value > 0.10

B                          0.05 < p-value < 0.10

C                          0.025 < p-value < 0.05

D                          0.01 < p-value < 0.025

E                           p-value < 0.01

c) Is the null hypothesis rejected at the 10% level of significance? No

C) Which of the following statements are true?

Note that there may be more than one correct answer; select all that are true.

1            The χ2 statistic can only take on values greater than or equal to 0.

4            When running a hypothesis test for standard deviation, if the sample size is large enough, our sample can come from a population with a non-Normal distribution

D) A random sample of size 12 is taken from a normally distributed population, and a sample variance of 16.0 is calculated.Use this information to test the null hypothesis H0:σ2=15 against the alternative hypothesis HA:σ2≠15.

Chi-Square Test of Variance
Data
Null Hypothesis                        s^2=	15
Level of Significance	0.05
Sample Size	12
Sample Standard Deviation	4
Intermediate Calculations
Degrees of Freedom	11
Half Area	0.025
Chi-Square Statistic	11.7333
Two-Tail Test
Lower Critical Value	3.8157
Upper Critical Value	21.9200
p-Value	0.3840
Do not reject the null hypothesis

a)What is the value of the test statistic χ2? Round your response to at least 3 decimal places.

test statistic χ2 = 11.733

b)The P-value falls within which one of the following ranges:

Answer: A P-value > 0.10

B 0.05 < P-value < 0.10

C 0.025 < P-value < 0.05

D 0.01 < P-value < 0.025

E            P-value < 0.01

c) What conclusion can be made, at the 5% level of significance?

Answer: A          There is insufficient evidence to reject the null hypothesis, and therefore no significant evidence that the population variance is not 15.