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 41 patients with arthritis took the drug for 3 months. Blood tests showed that x = 8.5 with sample standard deviation s = 3.2. 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 standard normal, since the sample size is large and σ is known. 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 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.

Answer 1

Answer: (a) For the given problem we construct the null and alternative hypotheses as:

H0: mu = 7.4 vs Ha: mu 7.4 [Since the claim was that, the drug has changed the pH level in either way so the test is two tailed]. mu = unknown true value of the parameter

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

The test statistic is T= (xbar-mu0)/(s/sqrt(n)) ; where xbar = sample mean, mu0 = the hypothesized value of the population mean, n = sample size, s = sample standard deviation, sqrt refers to the square root function. Under H0, T ~ t(n-1)

We reject H0 if p-value for T(observed) is less than alpha = level of significance.

Here, T(observed) = 2.201 (Rounded to three decimal places.)

(c) The p-value is = 2*P(x > 2.201)= 0.03357, so it is in the range < 0.0050.

The p-value here is the probability for obtaining a value less than 2.201 or greater than it. It is therefore an area under a t_n-1 curve to the left of -2.201 and right of 2.201 in a symmetrical set up.

The sketch is attached below:

(d) (d) Based on the answers in parts (a) to (c), at the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant, since p-value is less than the level of significance.

(e) Conclusion in the context of the application is that -- There is sufficient evidence at the 0.05 level to conclude that the drug has changed the mean pH level of the blood.