Calculating Z-Test Statistic for Hypothesis Testing

In a testing scenario, a researcher is conducting a hypothesis test to evaluate whether the mean weight of a certain population of fish is different from 4 pounds. The sample taken consists of 50 fish, revealing a sample mean weight of 4.5 pounds, with a known population standard deviation of 1 pound.

For conducting a two-tailed test with a significance level of $rac{1}{10}$ (0.1), calculate the appropriate test statistic used in this hypothesis test. The null and alternative hypotheses are defined as:

Null Hypothesis ($H_0$): The mean weight is equal to 4 pounds ($eta = 4$).

Alternative Hypothesis ($H_a$): The mean weight is not equal to 4 pounds ($eta eq 4$).

Use the formula for calculating the z-test statistic:

$$ z = rac{ar{x} - eta}{rac{ au}{ ext{sqrt}(n)}} $$

Where:

• $$ ar{x} $$ = sample mean
• $$ eta $$ = hypothesized population mean
• $$ au $$ = population standard deviation
• $$ n $$ = sample size