Question
Download Solution PDFSuppose that a sample of 100 independent draws from a normal distribution having unknown mean μ and known variance σ2 = 1 is observed. If the sample mean is 5, then the 95% confidence interval for μ is:
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDF- In this scenario, we have a sample of 100 independent draws from a normal distribution with an unknown mean μ and a known variance σ^2 = 1. The sample mean is given as 5, which is the average of the 100 observed values.
- To estimate the population mean μ, we can construct a confidence interval. A 95% confidence interval means that if we were to take many random samples from the same population and calculate confidence intervals using the same method, approximately 95% of those intervals would contain the true population mean.
- The confidence interval (4.804, 5.196) represents the range of values that are estimated to contain the true population mean μ with 95% confidence. The lower bound of 4.804 indicates the lower end of the range, and the upper bound of 5.196 indicates the upper end of the range.
- Therefore, based on the given sample data and the properties of the normal distribution, we can be 95% confident that the true population mean μ falls within the interval (4.804, 5.196).
- This means that if we were to repeat the sampling process and calculate a new confidence interval each time, approximately 95% of those intervals would contain the unknown true population mean μ.
Additional Information
- (4.84, 6.196): The upper bound in this option is 6.196, which is outside of the plausible range for the population mean given the sample mean of 5. This interval is too wide and would lead to a wider range of values than what is expected with 95% confidence.
- (4.23, 5.19): The lower bound in this option is 4.23, which is lower than the plausible range for the population mean given the sample mean of 5. This interval is too narrow and would exclude potential values within the plausible range of the population mean with 95% confidence.
- (4.804, 6.196): The upper bound in this option is 6.196, which is outside of the plausible range for the population mean given the sample mean of 5. This interval is too wide and would lead to a wider range of values than what is expected with 95% confidence.
Hence, If the sample mean is 5, then the 95% confidence interval for μ is (4.804, 5.196).
Last updated on Jul 21, 2025
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