Question

Given the following data set : 35, 29, 47, 39, 50, 38, and 41.

Construct a 99% confidence interval for the population mean.

Answer 1

The 99% confidence interval for the population mean is (32.45, 47.85), calculated using the sample data {35, 29, 47, 39, 50, 38, 41}. This interval suggests a high level of confidence that the true population mean falls within this range. The calculation involves the sample mean, standard deviation, and the Z-score corresponding to the desired confidence level.

Explanation:

The 99% confidence interval for the population mean, derived from the given dataset {35, 29, 47, 39, 50, 38, 41}, is (32.45, 47.85). To obtain this interval, the formula for a confidence interval is employed:
`\(\bar{x} \pm Z \left( (s)/(√(n)) \right)\)`, where
`\(\bar{x}\)` represents the sample mean, (s) is the sample standard deviation, (n) is the sample size, and (Z) is the Z-score corresponding to the desired confidence level.

For this calculation, the sample mean
`(\(\bar{x}\))` is 39, the sample standard deviation
`(\(s\))` is 6.07, and the sample size
`(\(n\))` is 7. The critical Z-score for a 99% confidence level is approximately 2.92. Substituting these values into the formula yields the confidence interval (32.45, 47.85). This interval suggests that we can be 99% confident that the true population mean falls within this range.

In practical terms, this means that if we were to draw numerous random samples from the same population and compute a 99% confidence interval for each, we would expect about 99% of these intervals to contain the true population mean. The width of the interval reflects the precision of our estimate, with a narrower interval indicating a more precise estimation of the population mean.