Question
A simple random sample of size 15 has mean x=41.8 and the population standard deviation is o=3.72. The population is normally distributed. Can you conclude that the population mean is less than 40?
The population standard deviation [is or isnt?] known.
The sample size n [is or isnt] greater than 30.
The population [is or is not?] approximately normal.
The correct decision is to [perform the z-test, perform the t-test or consult to a statistician?]
The population standard deviation [is or isnt?] known.
The sample size n [is or isnt] greater than 30.
The population [is or is not?] approximately normal.
The correct decision is to [perform the z-test, perform the t-test or consult to a statistician?]
Answers
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Answered by
GPT-4o mini
Given the information you provided, here are the answers to your questions:
1. **The population standard deviation is:** known (σ = 3.72).
2. **The sample size n:** is not greater than 30 (n = 15).
3. **The population:** is approximately normal (as stated).
Since the population standard deviation is known and the sample size is less than 30, we would generally use the **z-test** for hypothesis testing. This is because the population is normally distributed.
To conclude whether the population mean is less than 40, you would typically perform a one-sample z-test:
1. **Null Hypothesis (H0):** μ ≥ 40
2. **Alternative Hypothesis (H1):** μ < 40
4. Calculate the z-value using the formula:
\[
z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}
\]
Where:
- \( \bar{x} = 41.8 \) (sample mean)
- \( \mu_0 = 40 \) (hypothesized mean)
- \( \sigma = 3.72 \) (population standard deviation)
- \( n = 15 \)
5. Calculate:
\[
z = \frac{41.8 - 40}{3.72 / \sqrt{15}} \approx \frac{1.8}{0.962} \approx 1.87
\]
Next, you would compare this z-value to the critical z-value for your chosen significance level (commonly α = 0.05) for a one-tailed test.
If the calculated z-value is greater than the critical z-value (which would normally be about -1.645 for α = 0.05), you would fail to reject the null hypothesis.
Thus, the correct decision is to **perform the z-test**.
In summary:
- The population standard deviation **is** known.
- The sample size n **is not** greater than 30.
- The population **is** approximately normal.
- The correct decision is to **perform the z-test**.
1. **The population standard deviation is:** known (σ = 3.72).
2. **The sample size n:** is not greater than 30 (n = 15).
3. **The population:** is approximately normal (as stated).
Since the population standard deviation is known and the sample size is less than 30, we would generally use the **z-test** for hypothesis testing. This is because the population is normally distributed.
To conclude whether the population mean is less than 40, you would typically perform a one-sample z-test:
1. **Null Hypothesis (H0):** μ ≥ 40
2. **Alternative Hypothesis (H1):** μ < 40
4. Calculate the z-value using the formula:
\[
z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}
\]
Where:
- \( \bar{x} = 41.8 \) (sample mean)
- \( \mu_0 = 40 \) (hypothesized mean)
- \( \sigma = 3.72 \) (population standard deviation)
- \( n = 15 \)
5. Calculate:
\[
z = \frac{41.8 - 40}{3.72 / \sqrt{15}} \approx \frac{1.8}{0.962} \approx 1.87
\]
Next, you would compare this z-value to the critical z-value for your chosen significance level (commonly α = 0.05) for a one-tailed test.
If the calculated z-value is greater than the critical z-value (which would normally be about -1.645 for α = 0.05), you would fail to reject the null hypothesis.
Thus, the correct decision is to **perform the z-test**.
In summary:
- The population standard deviation **is** known.
- The sample size n **is not** greater than 30.
- The population **is** approximately normal.
- The correct decision is to **perform the z-test**.
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