Question

Consider the following hypotheses: H0: μ ≥ 150 HA: μ < 150

A sample of 80 observations results in a sample mean of 144. The population standard deviation is known to be 28. (You may find it useful to reference the appropriate table: z table or t table)

Calculate the value of the test statistic. (Negative value should be indicated by a minus sign. Round intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)_____

Answer 1

To calculate the test statistic for the hypothesis test involving a single population mean, the test statistic formula, Z = (μ - μ0)/( σ / √ n), is used, resulting in a Z score of -2.14.

Step-by-step explanation:

The student is asked to calculate the value of the test statistic for a hypothesis involving a single population mean. Given that the population standard deviation is known, a Z-test is appropriate for this hypothesis testing scenario. The null hypothesis, H0: μ ≥ 150, suggests the population mean is greater than or equal to 150, while the alternative hypothesis, HA: μ < 150, suggests the population mean is less than 150. With a sample mean of 144, a population standard deviation of 28, and a sample size of 80, the test statistic can be calculated using the formula:

Z = (μ - μ0)/( σ / √ n)

where Z is the test statistic, μ is the sample mean, μ0 is the mean under the null hypothesis, σ is the population standard deviation, and n is the sample size.

Plugging in the numbers:

Z = (144 - 150) / (28 / √ 80)

Performing the calculations:

Z = -2.1429

The calculated Z value, rounding to two decimal places, is -2.14. This is the test statistic that will be compared against the critical value from the Z table to determine if the null hypothesis can be rejected.