Identifying Hypothesis Test Parameters

The problem asks us to calculate the test statistic for a claim about the average weight of students. We are given information about the population standard deviation and a sample of students. This scenario calls for a z-test calculation.

Summary of Given Information:

• Claimed population mean weight ($\mu_0$): 82 kg
• Population standard deviation ($\sigma$): 20 kg
• Sample size ($n$): 81 students
• Sample mean weight ($\bar{x}$): 90 kg

Calculating the Z-Test Statistic

The test statistic helps determine how far the sample mean is from the hypothesized population mean, measured in terms of standard errors. Since the population standard deviation ($\sigma$) is known and the sample size ($n=81$) is large (greater than or equal to 30), we use the z-statistic formula:

$$z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$$

Step-by-Step Calculation Process:

1. Calculate the Standard Error (SE): The standard error of the mean estimates the variability of sample means.

   The formula is: $$ \text{SE} = \frac{\sigma}{\sqrt{n}} $$

   Plugging in the values: $$ \text{SE} = \frac{20}{\sqrt{81}} $$

   Since $\sqrt{81} = 9$: $$ \text{SE} = \frac{20}{9} $$

2. Calculate the z-statistic: Use the sample mean, claimed population mean, and the calculated standard error.

   The formula is: $$ z = \frac{\bar{x} - \mu_0}{\text{SE}} $$

   Substituting the values: $$ z = \frac{90 - 82}{\frac{20}{9}} $$

   Calculate the difference in the numerator: $$ z = \frac{8}{\frac{20}{9}} $$

   To divide by a fraction, multiply by its reciprocal: $$ z = 8 \times \frac{9}{20} $$

   Perform the multiplication: $$ z = \frac{72}{20} $$

   Simplify the fraction: $$ z = 3.6 $$

Final Test Statistic Value

The calculation shows that the test statistic is 3.6. This result is derived directly from the provided sample data and the instructor's claim about the population mean weight.