We want to conduct a hypothesis test of the claim that the population mean reading speed of second graders is different from 30.2 words per minute. So, we choose a random sample of students' reading speeds. The sample has a mean of 30.7 words per minute and a standard deviation of 3 words per minute, For each of the following sampling scenarios, choose an appropriate test statistic for our hypothesis test on the population mean. Then calculate that statistic. Round your answers to two decimal places. (a) The sample has size 100, and it is from a non-normally distributed population with a known standard deviation of 3. \[ =\square \] It is unclear which test statistic to use. (b) The sample has size 11, and it is from a normally distributed population with an unknown standard deviation. \( z=\square \) \( i=\square \)

Let's analyze each scenario step-by-step to determine the appropriate test statistic and calculate its value. ### **Given Information:** - **Hypothesis Test:** - Null Hypothesis (\( H_0 \)): \( \mu = 30.2 \) words per minute - Alternative Hypothesis (\( H_A \)): \( \mu \neq 30.2 \) words per minute - **Sample Statistics:** - Sample Mean (\( \bar{x} \)) = 30.7 words per minute - Sample Standard Deviation (\( s \)) = 3 words per minute ### **Part (a):** **Scenario:** - **Sample Size (\( n \))** = 100 - **Population Distribution:** Non-normally distributed - **Population Standard Deviation (\( \sigma \))** = 3 (known) **Choosing the Appropriate Test Statistic:** - **Large Sample Size (\( n = 100 \))**: By the Central Limit Theorem, the sampling distribution of the sample mean is approximately normal, regardless of the population distribution. - **Known Population Standard Deviation (\( \sigma \))**: Allows the use of the **Z-test**. **Calculation:** \[ z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} = \frac{30.7 - 30.2}{\frac{3}{\sqrt{100}}} = \frac{0.5}{0.3} \approx 1.67 \] **Answer:** \[ \boxed{z = 1.67} \] --- ### **Part (b):** **Scenario:** - **Sample Size (\( n \))** = 11 - **Population Distribution:** Normally distributed - **Population Standard Deviation (\( \sigma \))** = Unknown **Choosing the Appropriate Test Statistic:** - **Small Sample Size (\( n = 11 \))**: The sampling distribution is only normal if the population is normal. - **Unknown Population Standard Deviation:** Requires the use of the **t-test**. **Calculation:** \[ t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} = \frac{30.7 - 30.2}{\frac{3}{\sqrt{11}}} \approx \frac{0.5}{0.9045} \approx 0.55 \] **Answers:** \[ z = \text{Not applicable} \] \[ \boxed{t = 0.55} \] --- ### **Summary:** - **(a)** Use the **Z-test** with \( z = 1.67 \). - **(b)** Use the **t-test** with \( t = 0.55 \). ### **Final Answers:** **(a)** \( z = 1.67 \) **(b)** \[ z = \text{Not applicable} \] \[ t = 0.55 \]