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:

CI=xˉ±zα/2×snCI = x̄ \pm z_{\alpha/2} \times \frac{s}{\sqrt{n}}

Where:

  • 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:

Margin of error=1.96×2582\text{Margin of error} = 1.96 \times \frac{25}{\sqrt{82}}

First, compute the standard error:

2582259.0552.763\frac{25}{\sqrt{82}} \approx \frac{25}{9.055} \approx 2.763

Then, calculate the margin of error:

1.96×2.7635.4171.96 \times 2.763 \approx 5.417

Now, calculate the confidence interval:

CI=250±5.417CI = 250 \pm 5.417

Thus, the 95% confidence interval for the average number of customers per day is:

CI=(2505.417,250+5.417)(244.583,255.417)CI = (250 - 5.417, 250 + 5.417) \approx (244.583, 255.417)

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:

Margin of error=2.576×25822.576×2.7637.12\text{Margin of error} = 2.576 \times \frac{25}{\sqrt{82}} \approx 2.576 \times 2.763 \approx 7.12

Now, calculate the confidence interval:

CI=250±7.12CI = 250 \pm 7.12

Thus, the 99% confidence interval for the average number of customers per day is:

CI=(2507.12,250+7.12)(242.88,257.12)CI = (250 - 7.12, 250 + 7.12) \approx (242.88, 257.12)


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:

  1. How do we determine which confidence interval is appropriate in real-world situations?
  2. Why is the z-score used instead of a t-score when calculating confidence intervals in this case?
  3. How does the sample size (n) affect the width of the confidence interval?
  4. How would we calculate the confidence interval if the population standard deviation was known instead of the sample standard deviation?
  5. 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