Question

The Mean Corporation would like to invest in the booming health food industry. It is considering the creation of a health-drink franchise called Goose Juice. The investment department of the Mean Corporation wants to investigate the feasibility of this venture by examining the profits of similar franchises. It believes that the venture will be feasible if an average annual profit of more than $85,000 can be expected from each Goose Juice that is opened. It is known that the annual profits earned by health-drink franchises has a population standard deviation of $8,700.

The Mean Corporation's statisticians would like to construct a hypothesis test for the population mean annual profit earned by health-drink franchises. A random sample of 50 franchises was chosen and their annual profit for the previous financial year was recorded. The mean annual profit for the sample was calculated as $82,441.

The hypotheses that were used by the statisticians were

Null hypothesis: population mean = 81,000, Alternative hypothesis: population mean > 81,000 and the test statistic was found to be 1.29 (to 2 decimal places).

Required:
Calculate the p-value for this test, giving your answer as a decimal to 4 decimal places.

Answer 1

The p-value is approximately 0.9015

Explanation:

The statistic parameters are;

The population mean, μ₀ = $85,000

The population standard deviation, σ = $8,700

The size of the sample, n = 50

The sample mean annual profit,
`\bar x` = $82,441

The null hypothesis, H₀: μ = 81,000

The alternative hypothesis, Hₐ: μ > 81,000

Given that n > 30, a z-test is used

The test statistic ≈ 1.29

Therefore, we have;

`z=\frac{\bar{x}-\mu_0 }{(\sigma )/(√(n))} = 1.29`

From the z-score, the p value for a test statistic of 1.29 = 0.90147 which is 0.9015 after rounding up to 4 decimal places