Confidence Intervals for Estimating VCR Cost

Explanation

The interval widths and the z variates differ according to the rule chosen. The standard error is the same regardless.

To determine the difference between the 95% and 98% confidence intervals for estimating the true cost of the VCR, let's first understand the concepts involved.

A confidence interval is a range of values within which the true value of a population parameter, in this case, the cost of the VCR, is likely to fall. It is constructed based on a sample and provides a measure of the uncertainty associated with the estimate.

The width of a confidence interval refers to the range between the upper and lower bounds of the interval. A wider interval indicates greater uncertainty, while a narrower interval suggests higher precision in estimating the parameter.

The Z-variates, also known as Z-scores, are values obtained from the standard normal distribution. They represent the number of standard deviations a given observation is from the mean. Z-variates are used to calculate the critical values required to construct confidence intervals.

Now, let's analyze the given information. The survey of 144 retail stores revealed that the particular brand and model of the VCR retails for $375 with a standard deviation of $20.

To construct a confidence interval, we need to determine the critical values associated with the desired level of confidence. For a normal distribution, Z-variates are used to find these critical values. The critical values correspond to the tails of the distribution, indicating how much area lies outside the confidence interval.

For a 95% confidence interval, we look for the critical values associated with the tails that cover 2.5% on each side (5% in total) of the distribution. Similarly, for a 98% confidence interval, we look for the critical values associated with the tails that cover 1% on each side (2% in total) of the distribution.

Since the sample size is relatively large (n = 144), we can assume that the distribution of the sample mean is approximately normal due to the Central Limit Theorem. With a large sample size, we can use the Z-distribution instead of the T-distribution.

The formula for constructing a confidence interval for the population mean (assuming a normal distribution) is:

Confidence Interval = sample mean ± (Z * standard error)

The standard error is calculated as the standard deviation divided by the square root of the sample size:

Standard Error = standard deviation / sqrt(sample size)

For both the 95% and 98% confidence intervals, the only difference lies in the critical Z-values.

For a 95% confidence level, the critical Z-value is approximately 1.96. This means that 95% of the area under the normal distribution lies within 1.96 standard deviations of the mean.

For a 98% confidence level, the critical Z-value is approximately 2.33. This means that 98% of the area under the normal distribution lies within 2.33 standard deviations of the mean.

To summarize, the difference between the 95% and 98% confidence intervals lies in the critical Z-values used to calculate the interval bounds. The interval widths themselves are determined by the standard error, which is a function of the sample size and standard deviation. Therefore, the correct answer is:

D. Both interval widths and Z-variates