The population parameters for the ACT are a mean of 18 and standard deviation of 6. Let’s say you got a sample of 9 people with a mean of 24. Based on the z-test formulas this would give you a standard error of 2 and a z-score of 3. Plot (draw and upload a picture of) a sampling distribution including the following pieces of information:

Label the mean of the distribution using the correct symbol and number.
Plot the means for -3, -2, -1, 0, 1, 2, and 3 standard deviations away from the mean. In other words, below the distribution, at the appropriate locations (for these standard deviation increments), write the means that would sit at those locations.
Below those means, indicate the z-score for each mean (i.e., -3, -2, -1, 0, 1, 2, 3).
Mark the location of the negative and positive z-critical (-1.96; 1.96).
Mark the location of your sample mean of 24.

Here is the sampling distribution plot based on your request:

1. The mean of the distribution (\(\mu\)) is labeled at 18.
2. The corresponding sample means for -3, -2, -1, 0, 1, 2, and 3 standard deviations from the mean are marked below the curve.
3. The z-scores are also indicated below each sample mean.
4. The critical z-values (-1.96 and 1.96) are shown with dotted lines, marking the region where results would be statistically significant.
5. The sample mean of 24 is marked with a dashed red line.

Would you like any further clarification or more details?

Here are 5 related questions to expand on this topic:
1. How does the z-score change if the sample mean increases to 26?
2. What is the probability of getting a sample mean of 24 or higher?
3. How does changing the sample size affect the standard error?
4. What is the significance level for a z-critical value of ±2.58?
5. How would the graph change if the population standard deviation were 8 instead of 6?

**Tip:** A z-score tells you how many standard errors a sample mean is from the population mean, helping assess whether the result is typical or unusual.

The population parameters for the ACT are a mean of 18 and standard deviation of 6. Let’s say you got a sample of 9 people with a mean of 24. Based on the z-test formulas this would give you a standard error of 2 and a z-score of 3. Plot (draw and upload a picture of) a sampling distribution including the following pieces of information: Label the mean of the distribution using the correct symbol and number. Plot the means for -3, -2, -1, 0, 1, 2, and 3 standard deviations away from the mean. In other words, below the distribution, at the appropriate locations (for these standard deviation increments), write the means that would sit at those locations. Below those means, indicate the z-score for each mean (i.e., -3, -2, -1, 0, 1, 2, 3). Mark the location of the negative and positive z-critical (-1.96; 1.96). Mark the location of your sample mean of 24.

Explore the sampling distribution of a sample mean with ACT data, given a population mean of 18 and a standard deviation of 6. Learn how to calculate the z-score and standard error for a sample size of 9 with a mean of 24, including a visual representation of standard deviations and z-critical values.

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