Question

Centron, a pharmaceutical company, produces an allergy relief medication called Breezia. The company claims that the population standard deviation of the duration of Breezia's allergy relief is 2.50 hours. A scientist with the FDA suspects, however, that the standard deviation of the duration of Breezia's allergy relief is greater than (less consistent than) what the company claims.

The scientist obtains a random sample of 24 adults with allergy sensitivities, measures the duration of Breezia's allergy relief for each person, and calculates the sample standard deviation to be 3.25 hours.

Is there sufficient evidence to conclude that the standard deviation of the duration of Breezia's allergy relief is greater than (less consistent than) what the company claims? Use an α = 0.01 level of significance. Note: A normal probability plot and box plot indicate that the duration of allergy relief is normally distributed.

Question 1

State the null and alternative hypotheses (using symbols).

Question 2

Explain what it would mean for the scientist's study to make a Type I Error.

Question 3

Verify the requirements for this hypothesis test. Briefly show that each requirement is satisfied.

Question 4

Test the hypothesis using either the classical (test statistic) approach or the P-value approach. Show all steps to your approach. Then state whether you reject H 0 or do not reject H 0.

Question 5

Is there sufficient evidence to conclude that the standard deviation of the duration of Breezia's allergy relief is greater than (less consistent than) what the company claims? State your conclusion in one or two sentences.

Answer 1

Solution :

1) The null and alternative hypotheses would be as follows :

hours

hours

2) Type 1 error is defined as follows :

The error committed by rejecting the null hypothesis, when actually the null hypothesis is true, is known as type 1 error.

For the study, if scientist makes a type 1 error, it means that he is concluding that the standard deviation of the duration of Breezia's allergy relief is greater than 2.50 hours, when actually the standard deviation of the duration of relief is 2.50 hours.

3) The requirements for the test are as follows :

1) The population of duration relief must be normally distributed.

2) The sample must be random sample.

3) The sample observations must be independent.

All the three conditions are satisfied. All three are given in the problem statement.

4) To test the hypothesis we shall use chi-square test for variance. The test statistic is given as follows :

Where, n is sample size, s is sample standard deviation and σ is hypothesized value of population standard deviation under H​​​​​​0.

We have, n = 24, s = 3.25 hours,  σ = 2.50 hours

The value of the test statistic is 38.87.

Degrees of freedom = (n - 1) = (24 - 1) = 23

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

p-value = P(χ2 > value of the test statistic)

p-value = P(χ2 > 38.87)

p-value = 0.0205

The p-value is 0.0205.

Significance level (α) = 0.01

(0.0205 > 0.01)

Since, p-value is greater than the significance level of 0.01, therefore we shall be fail to reject the null hypothesis (H​​​​​​0) at 0.01 significance level.

Do not reject H​​​​​​0.

5) Conclusion : At 0.01 significance level, there is not sufficient evidence to conclude that the standard deviation of the duration of Breezia's allergy relief is greater than (less consistent than) what the company claims.

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