Math Problem Statement
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?
Solution
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:
-
Identify the given values:
- Population mean () = $75
- Population standard deviation () = $6
- Sample size () = 39
- We need to find the probability that the sample mean is between $76 and $77.
-
Calculate the standard error of the mean (SEM):
- Standardize the values (convert them to z-scores):
The z-score formula for the sample mean is:
We will compute the z-scores for $76 and $77:
- 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 .
- The z-score for $77 is approximately .
The probability that the sample mean of sales per customer is between $76 and $77 is approximately or .
Would you like any further details or assistance with related questions?
Related Questions
- How does the Central Limit Theorem apply to the distribution of the sample mean in this context?
- What impact does the sample size have on the standard error and the width of the confidence interval?
- How would the probability change if the sample size were larger or smaller?
- What is the interpretation of the z-score in this context?
- How can the store manager use this probability to make decisions about sales strategies?
- What are the assumptions underlying the use of the z-score for this calculation?
- How would you calculate the confidence interval for the sample mean?
- 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.
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Math Problem Analysis
Mathematical Concepts
Statistics
Sampling Distribution
Normal Distribution
Formulas
Standard Error of the Mean (SEM)
Z-score formula
Theorems
Central Limit Theorem
Suitable Grade Level
Advanced High School / College
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