Math Problem Statement
4/ Over a period of 82 days, you observed the number of customers and found that the average number of customers per day was 250people with s = 25.
a/ Calculate a 95% confidence interval for the average number of customers per day.
b/ Calculate a 99% confidence interval for the average number of customers per day
Solution
Given Information:
- Sample mean (x̄) = 250 customers/day
- Standard deviation (s) = 25 customers/day
- Sample size (n) = 82 days
Part a) 95% Confidence Interval:
To calculate the confidence interval, we use the formula:
Where:
- x̄ is the sample mean (250 customers)
- z_{\alpha/2} is the critical value for the desired confidence level
- s is the sample standard deviation (25 customers)
- n is the sample size (82 days)
For a 95% confidence level, the z_{\alpha/2} value is approximately 1.96 (you can find this from a standard normal distribution table).
Now, let's calculate the margin of error:
First, compute the standard error:
Then, calculate the margin of error:
Now, calculate the confidence interval:
Thus, the 95% confidence interval for the average number of customers per day is:
Part b) 99% Confidence Interval:
For a 99% confidence level, the z_{\alpha/2} value is approximately 2.576.
Again, calculate the margin of error:
Now, calculate the confidence interval:
Thus, the 99% confidence interval for the average number of customers per day is:
Final Answers:
- 95% Confidence Interval: (244.583, 255.417) customers per day
- 99% Confidence Interval: (242.88, 257.12) customers per day
Let me know if you'd like further details on these calculations!
Follow-up Questions:
- How do we determine which confidence interval is appropriate in real-world situations?
- Why is the z-score used instead of a t-score when calculating confidence intervals in this case?
- How does the sample size (n) affect the width of the confidence interval?
- How would we calculate the confidence interval if the population standard deviation was known instead of the sample standard deviation?
- How can confidence intervals be used to make decisions about customer service or marketing?
Tip: The larger the sample size, the more precise your confidence interval will be, and the narrower it will become.
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Math Problem Analysis
Mathematical Concepts
Statistics
Confidence Intervals
Normal Distribution
Formulas
Confidence Interval Formula: CI = x̄ ± z_{α/2} × (s / √n)
Margin of Error Formula: Margin of error = z_{α/2} × (s / √n)
Theorems
Central Limit Theorem
Z-distribution for Confidence Intervals
Suitable Grade Level
Grades 11-12
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