A random sample is drawn from a normally distributed...

A random sample is drawn from a normally distributed population of known standard deviation 10.7 . Construct a $95 \%$ confidence interval for the population mean based on the information given (not all of the information given need be used). a. $n=25, \bar{x}-100.3, s=11.0$ b. $n=4, \bar{x}-100.3,5=11.0$

A random sample is drawn from a normally distributed population of known standard deviation 10.7 . Construct a $95 \%$ confidence interval for the population mean based on the information given (not all of the information given need be used).
a. $n=25, \bar{x}-100.3, s=11.0$
b. $n=4, \bar{x}-100.3,5=11.0$

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the value of an unknown population parameter. It is constructed using the sample mean along with a margin of error, which is determined by the chosen confidence level and the variability in the population. In the context of normally distributed populations, a confidence interval for the mean provides an interval estimate that, across many samples, would capture the true population mean a specified percentage of the time (in this case, 95%).

Known Population Standard Deviation

When the standard deviation of the population is known, the construction of the confidence interval for the mean uses the z-distribution rather than the t-distribution. This is because the z-distribution applies when the variability of the population is fixed and known, allowing for the use of standard normal critical values to determine the margin of error.

Z-Distribution and Critical Values

The z-distribution is a standard normal distribution used in the construction of confidence intervals when the population standard deviation is known and the sample size is sufficiently large or the population is normally distributed. For a 95% confidence level, the critical value from the z-distribution is approximately 1.96, which is multiplied by the standard error to determine the margin of error.

Margin of Error

The margin of error in a confidence interval quantifies the amount of random sampling error in the estimate of a population parameter. It is calculated by multiplying the critical value (from the z-distribution for known population standard deviation) by the standard error (the population standard deviation divided by the square root of the sample size). This margin is then added to and subtracted from the sample mean to form the interval.

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