A personnel manager has found that historically the scores...

A personnel manager has found that historically the scores on aptitude tests given to applicants for entry level positions follow a normal distribution with a standard deviation of 32.4 points. A random sample of nine test scores from the current group of applicants had a mean score of 187.9 points. a. Find an $80 \%$ confidence interval for the population mean score of the current group of applicants. b. Based on these sample results, a statistician found for the population mean a confidence interval extending from 165.8 to 210.0 points. Find the confidence level of this interval.

A personnel manager has found that historically the scores on aptitude tests given to applicants for entry level positions follow a normal distribution with a standard deviation of 32.4 points. A random sample of nine test scores from the current group of applicants had a mean score of 187.9 points.

a. Find an $80 \%$ confidence interval for the population mean score of the current group of applicants.
b. Based on these sample results, a statistician found for the population mean a confidence interval extending from 165.8 to 210.0 points. Find the confidence level of this interval.

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about its mean, with data near the mean more frequent in occurrence than data far from the mean. It is commonly used in inferential statistics, especially when the population data are assumed to be distributed normally, allowing the use of standard techniques based on its properties.

Confidence Interval

A confidence interval is an estimated range of values which is likely to include an unknown population parameter. It is constructed from sample data along with a critical value that corresponds to the desired level of confidence, reflecting the degree of uncertainty in the estimation process.

Standard Error

The standard error measures the variability of a sample statistic, particularly the sample mean, by quantifying the dispersion of its sampling distribution. When the population standard deviation is known, the standard error is computed by dividing the population standard deviation by the square root of the sample size, playing a crucial role in constructing confidence intervals.

Z Distribution

The z distribution, or standard normal distribution, is used when the population standard deviation is known and the underlying distribution is normal. It provides critical values (z-scores) that are used to determine the margin of error in confidence interval calculations and hypothesis testing.

Critical Value

A critical value is a point on the scale of the test statistic beyond which we reject the null hypothesis, or that defines the limits of a confidence interval. In the context of confidence intervals for a normally distributed population, the critical value is obtained from the z distribution corresponding to the chosen confidence level.

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