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 36 patients with arthritis took the drug for 3 months. Blood tests showed that x = 8.6 with sample standard deviation s = 3.5. 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) State the null and alternate hypotheses.

(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 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.

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.)

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

(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.    

Answer 1

a) Here, as we are testing the claim that the drug has changed (either way) the mean pH level of the blood, therefore the null and the alternate hypothesis here are given as:

b) As we are not given the population standard deviation, but the sample size is greater than 30, we may use the standard normal distribution here.

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

c) The test statistic value here is computed as:

Therefore 2.057 is the required test statistic value here.

As this is a two tailed test, the p-value is computed from the standard normal tables as:

p = 2P(Z > 2.0571) = 2*0.0198 = 0.0396

0.010 < P-value < 0.050 is the required interval here.

d) The p-value here is 0.0396 < 0.05 which is the level of significance, therefore the test is significant and we can reject the null hypothesis and conclude that the data is significant

e) As the data is significant and we rejected the null hypothesis, we can conclude that There is sufficient evidence at the 0.05 level to conclude that the drug has changed the mean pH level of the blood.