Question

Data on fifth-grade test scores (reading and mathematics) for 422 school districts in California yield Yˉ =678.5 and standard deviation SY=20.5. The 95% confidence interval for the mean test score in the population is ( , ). (Round your responses to two decimal places.)

Answer 1

The 95% confidence interval for the mean test score in the population is (675.67, 681.33).

Explanation:

To compute the confidence interval for the mean test score, given a sample mean (Y-bar) of 678.5 and a standard deviation (SY) of 20.5 for 422 school districts in California, we use the formula for a confidence interval:
`\(\text{CI} = \bar{Y} \pm Z * (SY)/(√(n))\),`where
`\(\bar{Y}\)`represents the sample mean,
`\(SY\)`is the standard deviation,
`\(n\)`is the sample size, and
`\(Z\)`is the critical value from the standard normal distribution corresponding to the desired confidence level.

For a 95% confidence level, the Z-value is approximately 1.96. Substituting the given values, the computation yields a confidence interval of (675.67, 681.33). This indicates that we are 95% confident that the true mean test score for the population of California school districts lies between 675.67 and 681.33.

This confidence interval suggests that the average test score for fifth-grade students in California falls within this range, considering the sample data. It's important to note that this interval provides a range of values within which the true population mean is likely to reside. This statistical method helps infer the population parameter from sample data, offering insights while acknowledging the inherent variability.