To calculate the confidence intervals for the mean number of rooms rented daily based on the given sample data, we will use the formula for the confidence interval for the mean when the population standard deviation is unknown, which employs the t-distribution. The formula is:
\[ \text{Confidence Interval} = \bar{x} \pm t_{\alpha/2} \left( \frac{s}{\sqrt{n}} \right) \]
where:
- \(\bar{x}\) = sample mean
- \(s\) = sample standard deviation
- \(n\) = sample size
- \(t_{\alpha/2}\) = t-distribution critical value for the desired confidence level and degrees of freedom (df = n - 1)
Given Data:
- Sample size, \(n = 25\)
- Sample mean, \(\bar{x} = 37\)
- Sample standard deviation, \(s = 24\)
6.1 Finding the 99% Confidence Interval
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Find the critical t-value for 99% confidence level:
- Confidence level = 99% → \(\alpha = 0.01\)
- \(df = n - 1 = 25 - 1 = 24\)
- Using a t-table or calculator to find \(t_{0.005}\) for 24 degrees of freedom:
- Approximate critical value \( t \approx 2.797\)
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Calculate the standard error (SE): \[ SE = \frac{s}{\sqrt{n}} = \frac{24}{\sqrt{25}} = \frac{24}{5} = 4.8 \]
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Calculate the margin of error (ME): \[ ME = t_{\alpha/2} \cdot SE = 2.797 \cdot 4.8 \approx 13.4376 \]
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Construct the confidence interval: \[ \text{CI}{99%} = \bar{x} \pm ME = 37 \pm 13.4376 \] \[ \text{CI}{99%} = (37 - 13.4376, 37 + 13.4376) = (23.5624, 50.4376) \]
6.2 Finding the 90% Confidence Interval
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Find the critical t-value for 90% confidence level:
- Confidence level = 90% → \(\alpha = 0.10\)
- \(df = 24\)
- Using a t-table or calculator to find \(t_{0.05}\) for 24 degrees of freedom:
- Approximate critical value \( t \approx 1.711\)
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Calculate the margin of error (ME): \[ ME = t_{\alpha/2} \cdot SE = 1.711 \cdot 4.8 \approx 8.2068 \]
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Construct the confidence interval: \[ \text{CI}{90%} = \bar{x} \pm ME = 37 \pm 8.2068 \] \[ \text{CI}{90%} = (37 - 8.2068, 37 + 8.2068) = (28.7932, 45.2068) \]
Final Results:
- 99% Confidence Interval: \((23.56, 50.44)\)
- 90% Confidence Interval: \((28.79, 45.21)\)
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