Question

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 31 patients with arthritis took the drug for 3 months. Blood tests showed that x = 8.7 with sample standard deviation s = 3.1. 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.

H0: μ = 7.4; H1: μ ≠ 7.4

H0: μ = 7.4; H1: μ > 7.4   

H0: μ = 7.4; H1: μ < 7.4

H0: μ > 7.4; H1: μ = 7.4

H0: μ ≠ 7.4; H1: μ = 7.4

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

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

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

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

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

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

(c) Estimate the P-value.

P-value > 0.250

0.100 < P-value < 0.250   

0.050 < P-value < 0.100

0.010 < P-value < 0.050

P-value < 0.010

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

(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?

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

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

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

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.

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

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

Bill Alther is a zoologist who studies Anna's hummingbird (Calypte anna).† Suppose that in a remote part of the Grand Canyon, a random sample of six of these birds was caught, weighed, and released. The weights (in grams) were as follows.

3.7 2.9 3.8 4.2 4.8 3.1

The sample mean is x = 3.75 grams. Let x be a random variable representing weights of hummingbirds in this part of the Grand Canyon. We assume that x has a normal distribution and σ = 0.66 gram. Suppose it is known that for the population of all Anna's hummingbirds, the mean weight is μ = 4.35 grams. Do the data indicate that the mean weight of these birds in this part of the Grand Canyon is less than 4.35 grams? Use α = 0.10.

(a) What is the level of significance?

State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test?

H0: μ = 4.35 g; H1: μ > 4.35 g; right-tailed

H0: μ < 4.35 g; H1: μ = 4.35 g; left-tailed   

H0: μ = 4.35 g; H1: μ ≠ 4.35 g; two-tailed

H0: μ = 4.35 g; H1: μ < 4.35 g; left-tailed

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

The standard normal, since we assume that x has a normal distribution with known σ.

The Student's t, since n is large with unknown σ.   

The standard normal, since we assume that x has a normal distribution with unknown σ.

The Student's t, since we assume that x has a normal distribution with known σ.

Compute the z value of the sample test statistic. (Round your answer to two decimal places.)

(c) Find (or estimate) the P-value. (Round your answer to four decimal places.)

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

(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?

At the α = 0.10 level, we reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.10 level, we reject the null hypothesis and conclude the data are not statistically significant.   

At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) State your conclusion in the context of the application.

There is sufficient evidence at the 0.10 level to conclude that humming birds in the Grand Canyon weigh less than 4.35 grams.

There is insufficient evidence at the 0.10 level to conclude that humming birds in the Grand Canyon weigh less than 4.35 grams.

Answer 1

Question 1

Part a)

α = 0.05

H0: μ = 7.4; H1: μ ≠ 7.4

part b)

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

Test Statistic :-
t = ( X̅ - µ ) / ( S / √(n))
t = ( 8.7 - 7.4 ) / ( 3.1 / √(31) )
t = 2.335

P - value = P ( t > 2.3349 ) = 0.0264

0.010 < P-value < 0.050

Decision based on P value
Reject null hypothesis if P value < α = 0.05 level of significance
P - value = 0.0264 < 0.05 ,hence we reject null hypothesis
Conclusion :- Reject null hypothesis

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

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

Question 2

Part a)

α = 0.10

H0: μ = 4.35 g; H1: μ < 4.35 g; left-tailed

part b)

The standard normal, since we assume that x has a normal distribution with known σ.

Test Statistic :-
Z = ( X - µ ) / ( σ / √(n))
Z = ( 3.75 - 4.35 ) / ( 0.66 / √( 6 ))
Z = -2.23

Part c)

P value = P ( Z < 2.2268 ) = 0.0130

Reject null hypothesis if P value < α = 0.1 level of significance
Since 0.013 < 0.1 ,hence we reject null hypothesis
Result :- Reject null hypothesis

At the α = 0.10 level, we reject the null hypothesis and conclude the data are statistically significant.

There is sufficient evidence at the 0.10 level to conclude that humming birds in the Grand Canyon weigh less than 4.35 grams.