Question

Two different formulations of an oxygenated motor fuel are being tested to study their road octane numbers. The population variance of road octane number for formulation 1 is =1.5, and for formulation 2, it is =1.2. Two random samples of size =15 and =20 are tested, and the mean road octane numbers observed are =89.6 and =92.5. Assume normality.

a.) Construct a 95% two-sided confidence interval on the difference in mean road octane number.

b.) If formulation 2 produces a higher road octane number than formulation 1, the manufacturer would like to detect it. Formulate and test an appropriate hypothesis, using α = 0.05.(Be sure to include all the steps).

c.) What is the P-value for the test you conducted in part (b) and state the decision?

Answer 1

a)

We need to construct the 95% confidence interval for the difference between the population means μ1​−μ2​. The following information has been provided about each of the samples:

Sample Mean 1 () =	89.6
Pop. Standard Deviation 1 (σ1​) =	1.224744871
Sample Size 1 (N1​) =	15
Sample Mean 2 (​) =	92.5
Pop. Standard Deviation 2 (σ2​) =	1.095445115
Sample Size 2 (N2​) =	20

The critical value for α=0.05 is . The corresponding confidence interval is computed as shown below:

b) and c)

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ1​=μ2​

Ha: μ1​<μ2​

This corresponds to a left-tailed test, for which a z-test for two population means, with known population standard deviations will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a left-tailed test is z_c = -1.64

The rejection region for this left-tailed test is R={z:z<−1.64}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that z=−7.25<z∗=−1.64, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0 and since p=0<0.05, it is concluded that the null hypothesis is rejected..

Graphically