Question

 

Q1)

A random sample of 36 values is drawn from a mound-shaped and symmetric distribution. The sample mean is 9 and the sample standard deviation is 2. Use a level of significance of 0.05 to conduct a two-tailed test of the claim that the population mean is 8.5.

(A) Is it appropriate to use a Student's t distribution? Explain.

a) Yes, because the x distribution is mound-shaped and symmetric and σ is unknown.

b) No, the x distribution is skewed left.

c) No, the x distribution is skewed right.

d) No, the x distribution is not symmetric.No, σ is known.

How many degrees of freedom do we use?


(B) What are the hypotheses?

a) H₀: μ < 8.5; H₁: μ = 8.5

b) H₀: μ = 8.5; H₁: μ ≠ 8.5

c) H₀: μ > 8.5; H₁: μ = 8.5

d) H₀: μ = 8.5; H₁: μ > 8.5

e) H₀: μ = 8.5; H₁: μ < 8.5

(C) Compute the t value of the sample test statistic. (Round your answer to three decimal places.)

t =

(D) Estimate the P-value for the test.

a) P-value > 0.250

b) 0.100 < P-value < 0.250

c) 0.050 < P-value < 0.100

d) 0.010 < P-value < 0.050

e) P-value < 0.010

(E) Do we reject or fail to reject H₀?

a) At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

b) At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.

c) At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

d) At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(F) Interpret the results.

a) There is sufficient evidence at the 0.05 level to reject the null hypothesis.

b) There is insufficient evidence at the 0.05 level to reject the null hypothesis.

Q2

Let x be a random variable that represents the pH of arterial plasma (i.e., acidity of the blood). For healthy adults, the mean of the x distribution is μ = 7.4.† A new drug for arthritis has been developed. However, it is thought that this drug may change blood pH. A random sample of 36 patients with arthritis took the drug for 3 months. Blood tests showed that x = 8.4 with sample standard deviation s = 2.7. Use a 5% level of significance to test the claim that the drug has changed (either way) the mean pH level of the blood.

(A) What is the level of significance?


State the null and alternate hypotheses.

a) H₀: μ ≠ 7.4; H₁: μ = 7.4

b) H₀: μ = 7.4; H₁: μ > 7.4    

c) H₀: μ = 7.4; H₁: μ < 7.4

d) H₀: μ = 7.4; H₁: μ ≠ 7.4

e) H₀: μ > 7.4; H₁: μ = 7.4

(B) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution.

a) The standard normal, since the sample size is large and σ is unknown.

b) The Student's t, since the sample size is large and σ is unknown.    

c) The standard normal, since the sample size is large and σ is known.

d) The Student's t, since the sample size is large and σ is known.

What is the value of the sample test statistic? (Round your answer to three decimal places.)


(C) Estimate the P-value.

a) P-value > 0.250

b) 0.100 < P-value < 0.250    

c) 0.050 < P-value < 0.100

d) 0.010 < P-value < 0.050

e) P-value < 0.010

Sketch the sampling distribution and show the area corresponding to the P-value.

Top left, bottom left, top right or bottom right?

(D) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

a) At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

b) At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.    

c) At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

d) At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(E) Interpret your conclusion in the context of the application.

a) There is sufficient evidence at the 0.05 level to conclude that the drug has changed the mean pH level of the blood.

b) There is insufficient evidence at the 0.05 level to conclude that the drug has changed the mean pH level of the blood.

Answer 1

1)

(A) Is it appropriate to use a Student's t distribution? Explain.

a) Yes, because the x distribution is mound-shaped and symmetric and σ is unknown.

How many degrees of freedom do we use?

df = n-1 = 36-1 = 35

(B) What are the hypotheses?

b) H0: μ = 8.5; H1: μ ≠ 8.5

(D) Estimate the P-value for the test.

a) P-value > 0.250

Here P-value is large.so we fail to reject the null hypothesis.

(E) Do we reject or fail to reject H0?

d) At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(F) Interpret the results.

b) There is insufficient evidence at the 0.05 level to reject the null hypothesis.

2)

(A) What is the level of significance?
0.05

State the null and alternate hypotheses.

d) H0: μ = 7.4; H1: μ ≠ 7.4

(B) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution.

b) The Student's t, since the sample size is large and σ is unknown.    

d) 0.010 < P-value < 0.050

P-value = 0.0328 < 0.05

So we reject Ho.

(D) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Is the data statistically significant at level α?

a) At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

(E) Interpret your conclusion in the context of the application.

a) There is sufficient evidence at the 0.05 level to conclude that the drug has changed the mean pH level of the blood.