The standard deviation for a population is σ = 15.3. A sample of 36 observations selected from this
population gave a mean equal to 74.8
a. Make a 90% confidence interval for µ
b. Construct a 95% confidence interval for µ
c. Determine a 99% confidence interval for µ
d. Does the width of the confidence intervals constructed in parts a through c increase as the confidence
level increases? Explain your answer�
To construct confidence intervals for the population mean (µ) based on a sample, you can use the formula:
Confidence Interval = sample mean ± (critical value * standard deviation / square root of sample size)
Where:
- Sample mean is the mean obtained from the sample observations.
- Critical value is the value obtained from the Z-table or T-table, depending on the sample size and the population standard deviation.
- Standard deviation (σ) is the measure of variability in the population.
- Sample size is the number of observations in the sample.
a. 90% Confidence Interval:
To calculate a 90% confidence interval, you need to find the critical value for a 90% confidence level. Since the sample size is greater than 30, you can use the Z-table. From the Z-table, the critical value for a 90% confidence level (two-tailed) is approximately 1.645.
90% Confidence Interval = 74.8 ± (1.645 * 15.3 / √36)
90% Confidence Interval = 74.8 ± (1.645 * 15.3 / 6)
90% Confidence Interval ≈ 74.8 ± 4.025
Lower Limit = 74.8 - 4.025 ≈ 70.775
Upper Limit = 74.8 + 4.025 ≈ 78.825
Therefore, the 90% confidence interval for µ is approximately (70.775, 78.825).
b. 95% Confidence Interval:
To calculate a 95% confidence interval, you need to find the critical value for a 95% confidence level. Since the sample size is greater than 30, you can still use the Z-table. From the Z-table, the critical value for a 95% confidence level (two-tailed) is approximately 1.96.
95% Confidence Interval = 74.8 ± (1.96 * 15.3 / √36)
95% Confidence Interval = 74.8 ± (1.96 * 15.3 / 6)
95% Confidence Interval ≈ 74.8 ± 4.7367
Lower Limit = 74.8 - 4.7367 ≈ 70.0633
Upper Limit = 74.8 + 4.7367 ≈ 79.5367
Therefore, the 95% confidence interval for µ is approximately (70.0633, 79.5367).
c. 99% Confidence Interval:
To calculate a 99% confidence interval, you still need to find the critical value. Since the sample size is greater than 30, you can again use the Z-table. From the Z-table, the critical value for a 99% confidence level (two-tailed) is approximately 2.576.
99% Confidence Interval = 74.8 ± (2.576 * 15.3 / √36)
99% Confidence Interval = 74.8 ± (2.576 * 15.3 / 6)
99% Confidence Interval ≈ 74.8 ± 6.6267
Lower Limit = 74.8 - 6.6267 ≈ 68.1733
Upper Limit = 74.8 + 6.6267 ≈ 81.4267
Therefore, the 99% confidence interval for µ is approximately (68.1733, 81.4267).
d. Yes, the width of the confidence intervals constructed in parts a through c increases as the confidence level increases. This is because as you increase the confidence level, you need to include a larger range of values to be confident about capturing the true population mean within the interval. As a result, the wider range leads to a wider confidence interval.