Question

Consider the following hypothesis test:
\(Η_0: μ ≤ 20\)
\(Η_1 : μ > 20\)
A sample of 81 produced a sample mean of 20.55. The population standard deviation is 3. The value of the test statistic is:

• A. 1.85
• B. -2.05
• C. -2.15
• D. 1.65

Answer 1

The Correct Option is D

To solve the hypothesis test problem and determine the value of the test statistic, follow these steps:

1. Define the hypotheses:

\( H_0: \mu \leq 20 \) (null hypothesis)
\( H_1: \mu > 20 \) (alternative hypothesis)

2. Given data:

• Sample size: \( n = 81 \)
• Sample mean: \( \bar{x} = 20.55 \)
• Population standard deviation: \( \sigma = 3 \)

3. Use the formula for the z-test statistic:

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

4. Substitute the values:

\( z = \frac{20.55 - 20}{\frac{3}{\sqrt{81}}} \)

5. Simplify:

• Denominator: \( \frac{3}{\sqrt{81}} = \frac{3}{9} = 0.33 \)
• Numerator: \( 20.55 - 20 = 0.55 \)

6. Calculate the test statistic:

\( z = \frac{0.55}{0.33} = 1.65 \)

\[ \boxed{\text{Test Statistic } z = 1.65} \]