It is known that the standard deviation in the volumes of...

It is known that the standard deviation in the volumes of 20-ounce (591-milliliter) bottles of natural spring water bottled by a particular company is 5 millliliters. One hundred bottles are randomly sampled and measured. a. Calculate the standard error of the mean. b. Find the margin of error of a $90 \%$ confidence interval estimate for the population mean volume. c. Calculate the width for a $98 \%$ confidence interval for the population mean volume.

It is known that the standard deviation in the volumes of 20-ounce (591-milliliter) bottles of natural spring water bottled by a particular company is 5 millliliters. One hundred bottles are randomly sampled and measured.
a. Calculate the standard error of the mean.
b. Find the margin of error of a $90 \%$ confidence interval estimate for the population mean volume.
c. Calculate the width for a $98 \%$ confidence interval for the population mean volume.

Key Concepts

Critical Value

The critical value is the point on the distribution (often the standard normal distribution) associated with a given level of confidence that defines the boundaries of the confidence interval. For example, common critical values are approximately 1.645 for a 90% confidence level and 2.33 for a 98% confidence level. The critical value is used to scale the standard error to form the margin of error.

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 specified level of confidence (e.g., 90% or 98%). The interval is constructed by taking the sample mean and adding and subtracting the margin of error. Understanding confidence intervals is essential for making inferences about population parameters from sample data.

Margin of Error

The margin of error represents the maximum expected difference between the sample statistic (such as the mean) and the true population parameter at a given level of confidence. It is calculated by multiplying the critical value (obtained from the standard normal or other relevant distribution) by the standard error. The margin of error helps quantify the uncertainty associated with estimating the true value of a parameter.

Standard Error

The standard error of the mean is the measure of how much variability is expected in the sample mean from sample to sample when drawn from the same population. It is calculated by dividing the population standard deviation by the square root of the sample size. This concept is fundamental in inferential statistics because it quantifies the precision of the sample mean as an estimate of the population mean.

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