A random sample of 90 observations produced a mean x̅=25.9 and a standard deviation s=2.7 . a. F

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A random sample of 90 observations produced a mean x̅=25.9...

A random sample of 90 observations produced a mean $\bar{x}=25.9$ and a standard deviation $s=2.7 .$
a. Find a $95 \%$ confidence interval for the population $\operatorname{mean} \mu .$
b. Find a $90 \%$ confidence interval for $\mu$.
c. Find a $99 \%$ confidence interval for $\mu$.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a certain level of confidence. It provides an estimate of uncertainty or variability in the measurement of the parameter, such as the population mean, and is expressed as the sample statistic plus or minus the margin of error.

Standard Error

The standard error measures the variability of a sample statistic, typically the sample mean, from the true population parameter. It is calculated as the sample standard deviation divided by the square root of the sample size. The standard error is essential for constructing confidence intervals as it quantifies the precision of the sample estimate.

Critical Value

The critical value is a point on the scale of the test statistic, such as a t-statistic or z-statistic, that corresponds to the desired confidence level. It determines the number of standard errors to extend above and below the sample mean to create a confidence interval. The critical value varies with the chosen level of confidence and the sample size, especially when using the t-distribution.

t-Distribution

The t-distribution is used in inferential statistics when the sample size is small or when the population standard deviation is unknown. It resembles the normal distribution but has heavier tails, providing a more conservative estimate for confidence intervals. The t-distribution is indexed by degrees of freedom, which are typically calculated as the sample size minus one.

Margin of Error

The margin of error represents the maximum expected difference between the true population parameter and the sample estimate. It is calculated by multiplying the critical value by the standard error. The margin of error provides a quantification of the uncertainty in the estimate and widens as the level of confidence increases or as variability in the data increases.

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