A simple random sample of size 33 has mean . The population standard deviation is σ=3.72. The population is normally distributed. Can you conclude that the population mean is greater than ?

Question

The population standard deviation [?] known.
The sample size n [?] greater than .
The correct decision is to [?]

Answer 1

To determine whether we can conclude that the population mean is greater than some value, we should perform a hypothesis test. Here are the general steps and details based on the information you provided:

1. Define the null and alternative hypotheses:

   • Null hypothesis (\(H_0\)): The population mean (\(\mu\)) is less than or equal to a specified value (let's denote it as \( \mu_0 \)).
   • Alternative hypothesis (\(H_a\)): The population mean (\(\mu\)) is greater than \( \mu_0 \).

2. Significance level: Assuming a common significance level (for example, \(\alpha = 0.05\)) for your test.

3. Calculate the test statistic:

   • Because the population standard deviation (\(\sigma = 3.72\)) is known and the sample size (\(n = 33\)) is greater than 30, you can use the Z-test for the hypothesis testing.
   • The Z statistic is calculated as: \[ Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \] where:
   • \(\bar{x}\) is the sample mean,
   • \(\sigma\) is the population standard deviation,
   • \(n\) is the sample size.

4. Determine the critical value:

   • Look up the Z-table for the critical value that corresponds to your significance level (\(\alpha\)). For \(\alpha = 0.05\), the critical value for a one-tailed test is approximately 1.645.

5. Decision rule:

   • If the calculated Z statistic is greater than the critical value (1.645 for \(\alpha = 0.05\)), you reject the null hypothesis. Otherwise, you fail to reject the null hypothesis.

6. Conclusion:

   • If you reject the null hypothesis, you conclude that there is sufficient evidence to claim that the population mean is greater than \( \mu_0 \).
   • If you fail to reject the null hypothesis, you conclude that there is not enough evidence to claim that the population mean is greater than \( \mu_0 \).

7. Fill in the blanks:

   • The population standard deviation: 3.72 (known).
   • The sample size: 33 (greater than 30).
   • The correct decision is to either: reject the null hypothesis or fail to reject the null hypothesis, depending on your calculations.

To complete the analysis, you would need the specific value of the sample mean (\(\bar{x}\)) and the value you are testing against (\( \mu_0 \)).

Answer 2

The correct decision is to [perform the z-test or t-test? or consult a statistician?]

Answer 3

Based on the information you've provided, you have a simple random sample, a known population standard deviation, and a sample size of 33. Since the population standard deviation is known and the sample size is sufficiently large (greater than 30), you should perform the Z-test.

In summary:

• Perform the Z-test: This is appropriate here since you have a known population standard deviation and your sample size is sufficiently large (n > 30).

The Z-test will allow you to determine if there is enough statistical evidence to conclude that the population mean is greater than the specified value (\( \mu_0 \)).

Consulting a statistician could be helpful if you need guidance interpreting results or if the context is particularly complex, but for this scenario, a Z-test is the correct statistical approach.

Answer 4

A simple random sample of size 13 has mean x=14.3 . The population standard deviation is σ=3.72. The population is not normally distributed. Can you conclude that the population mean is less than 10?

The population standard deviation [is/isnt?] known.
The sample size n [is/isnt?] greater than .
The correct decision is to [perform the z-test or t-test? or consult a statistician?]

Answer 5

In your scenario, you have a simple random sample of size 13 with a sample mean \(\bar{x} = 14.3\) and a known population standard deviation \(\sigma = 3.72\). You also mentioned that the population is not normally distributed and you want to test if the population mean is less than 10.

Let's evaluate each part:

1. The population standard deviation:

   • is known: You have provided the population standard deviation (\(\sigma = 3.72\)).

2. The sample size \(n\):

   • is not greater than 30: The sample size is 13, which is less than 30.

3. The correct decision:

   • Since the sample size is small (n < 30) and you mentioned that the population is not normally distributed, it is not appropriate to use the Z-test since the assumption of normality is violated, even with a known population standard deviation.
   • Therefore, you should perform the t-test instead. The t-test is more robust for small sample sizes, especially when the underlying population is not normally distributed.

In summary:

• The population standard deviation is known.
• The sample size \(n\) is not greater than 30.
• The correct decision is to perform the t-test.

Answer 6

A simple random sample of size 28 has mean x=41.8. The population standard deviation is s=1.9. The population is normally distributed. Can you conclude that the population mean differs from 40?

The population standard deviation [is/isnt?] known.
The sample size n [is/isnt?] greater than 30.
The correct decision is to [perform the z-test or t-test? or consult a statistician?]

Answer 7

In your scenario, you have a simple random sample of size 28 with a sample mean \(\bar{x} = 41.8\) and a sample standard deviation \(s = 1.9\). The population is stated to be normally distributed, and you want to test if the population mean differs from 40.

Let’s evaluate each part:

1. The population standard deviation:

   • isn't known: Although you have the sample standard deviation (\(s = 1.9\)), the population standard deviation is not provided. Therefore, we treat the sample standard deviation as an estimate of the population standard deviation.

2. The sample size \(n\):

   • isn't greater than 30: The sample size is 28, which is less than 30.

3. The correct decision:

   • Since the sample size is less than 30 and the population standard deviation is unknown (we're using the sample standard deviation instead), it's appropriate to use the t-test for this hypothesis testing.

Summary:

• The population standard deviation isn't known.
• The sample size \(n\) isn't greater than 30.
• The correct decision is to perform the t-test.

Hypothesis Testing Steps:

1. Set up the hypotheses:

   • Null Hypothesis (\(H_0\)): \(\mu = 40\) (the population mean is 40).
   • Alternative Hypothesis (\(H_a\)): \(\mu \neq 40\) (the population mean differs from 40).

2. Determine the significance level (e.g., \(\alpha = 0.05\)).

3. Calculate the t-statistic using the formula: \[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \] where \(\bar{x} = 41.8\), \(\mu_0 = 40\), \(s = 1.9\), and \(n = 28\).

4. Find the critical t-value from the t-distribution table based on \(n - 1 = 27\) degrees of freedom for your chosen \(\alpha\).

5. Make a decision based on the calculated t-statistic and the critical t-value. If the absolute value of the t-statistic is greater than the critical value, you will reject the null hypothesis. If not, you will fail to reject the null hypothesis.

These steps will help you conclude whether there is sufficient evidence to suggest that the population mean differs from 40.