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 = 8.1 with sample standard deviation s = 1.9. Use a 5% level of significance to test the claim that the drug has changed (either way) the mean pH level of the blood

Answer 1

Solution :

The null and alternative hypotheses are as follows:

H​​​​​​​​​​​0 : μ = 7.4 i.e. The population mean of the blood pH of all arthritis patients who took the new drug is 7.4.

H​​​​​​​​​​1 : μ ≠ 7.4 i.e. The population mean of the blood pH of all arthritis patients who took the new drug is not equal to 7.4.

To test the hypothesis we shall use one sample t-test. The test statistic is given as follows :

Where, x̄ is sample mean, μ is hypothesized value of population mean, s is sample standard deviation and n is sample size.

We have, x̄ = 8.1, s = 1.9,  μ = 7.4 and n = 31

The value of the test statistic is 2.0513.

Since, our test is two-tailed test, therefore we shall obtain two-tailed p-value for the test statistic, which is given as follows:

p-value = 2.P(T > t)

p-value = 2.P(T > 2.0513)

p-value = 0.04906

Significance level = 5% = 0.05

(0.04906 < 0.05)

Since, p-value is less than the significance level of 5%, therefore we shall reject the null hypothesis (H​​​​​​0) at significance level of 5%.

Conclusion : At 5% significance level, there is enough evidence to support the claim that the drug has changed (either way) the mean pH level of the blood.

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