According to the latest financial reports from a sporting goods store, the mean sales per customer was $75 with a population standard deviation of $6. The store manager believes 39 randomly selected customers spent more per transaction.

What is the probability that the sample mean of sales per customer is between $76 and $77 dollars?


To determine the probability that the sample mean of sales per customer is between $76 and $77, given that the population mean sales is $75 with a population standard deviation of $6, and the sample size is 39, we can use the sampling distribution of the sample mean.

### Steps to Solve the Problem:

1. **Identify the given values:**
   - Population mean ($\mu$) = $75
   - Population standard deviation ($\sigma$) = $6
   - Sample size ($n$) = 39
   - We need to find the probability that the sample mean $\bar{x}$ is between $76 and $77.

2. **Calculate the standard error of the mean (SEM):**

\[
\text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{6}{\sqrt{39}}
\]

3. **Standardize the values (convert them to z-scores):**

The z-score formula for the sample mean is:

\[
z = \frac{\bar{x} - \mu}{\text{SEM}}
\]

We will compute the z-scores for $76 and $77:

\[
z_{76} = \frac{76 - 75}{\text{SEM}}
\]
\[
z_{77} = \frac{77 - 75}{\text{SEM}}
\]

4. **Use the standard normal distribution to find the probabilities corresponding to these z-scores.**

Let's proceed with these calculations.The calculations yield the following results:

- The z-score for $76 is approximately $ z \approx 1.04 $.
- The z-score for $77 is approximately $ z \approx 2.08 $.

The probability that the sample mean of sales per customer is between $76 and $77 is approximately $ 0.1303 $ or $ 13.03\% $.

Would you like any further details or assistance with related questions?

### Related Questions

1. How does the Central Limit Theorem apply to the distribution of the sample mean in this context?
2. What impact does the sample size have on the standard error and the width of the confidence interval?
3. How would the probability change if the sample size were larger or smaller?
4. What is the interpretation of the z-score in this context?
5. How can the store manager use this probability to make decisions about sales strategies?
6. What are the assumptions underlying the use of the z-score for this calculation?
7. How would you calculate the confidence interval for the sample mean?
8. What other statistical tests could be used to analyze the sales data?

### Tip

When dealing with probabilities and z-scores, remember that the standard normal distribution is symmetrical around the mean, and probabilities can be easily looked up in standard normal tables or calculated using statistical software.

Calculate the probability that the sample mean of sales per customer is between $76 and $77 using statistical methods. Understand the implications of population mean, standard deviation, and sample size in probability calculations. Learn how to apply the Central Limit Theorem and z-score calculations in real-world scenarios.

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