Question

How much money do people spend on graduation gifts? In \(2007,\) the National Retail Federation (www.nrf.com) surveyed 2,815 consumers who reported that they bought one or more graduation gifts that year. The sample was selected to be representative of adult Americans who purchased graduation gifts in 2007 . For this sample, the mean amount spent per gift was \(\$ 55.05\). Suppose that the sample standard deviation was \$20. Construct and interpret a \(98 \%\) confidence interval for the mean amount of money spent per graduation gift in 2007 .

Short answer

To construct a 98% confidence interval for the mean amount spent on graduation gifts in 2007, first calculate the standard error using the given sample size and standard deviation. Then look up the critical value associated with a .01 level of significance in a z-distribution table (which is roughly 2.33). Finally, multiply the critical value by the standard error and add and subtract this quantity from the sample mean. The result is the 98% confidence interval.

Step-by-step solution

Calculation of Standard Error

The standard error (SE) here is calculated using the formula SE = \(\dfrac{sample Standard Deviation}{\sqrt{sample Size}}\). The given standard deviation is $20 and the size is 2815. Therefore, the computation is as follows: SE = \(\dfrac{20}{\sqrt{2815}}\)

Determination of the critical value

First, determine the level of significance (\(\alpha\)) by subtracting the confidence level from 1, which gives \(\alpha = 1 - \dfrac{98}{100} = 0.02\). The critical value is found from a z-table which corresponds to \(\alpha/2 = 0.01\), and this is approximately 2.33.

Construction of the confidence interval

The confidence interval (CI) is calculated using the formula CI = sample mean \u00B1 (Critical value * SE). Therefore the confidence interval is $55.05 \u00B1 2.33 * SE.

Key concepts

Standard Error Calculation

Understanding how to calculate the standard error (SE) is crucial for statistical analysis, especially when constructing confidence intervals. The standard error measures the amount of variation or 'error' to expect in a sample mean, compared to the true population mean. It's a way of seeing how good our sample is at estimating the actual mean if we were able to survey the whole population.

The formula to calculate SE is quite straightforward: SE = \(\frac{sample \; Standard \; Deviation}{\sqrt{sample \; Size}}\). Take the sample standard deviation, which tells us how spread out the individual sample values are from the sample mean, and divide it by the square root of the total number of individuals in that sample, referred to as the sample size. This calculation gives us insight into how accurate our sample mean is as an estimate of the population mean. Lower standard errors mean the sample mean is a more accurate estimate.

Determination of Critical Value

When constructing confidence intervals, determining the critical value is a step that cannot be overlooked. The critical value is a point on the distribution curve that defines the margins for a given confidence level. It's essentially the cut-off value where we say that our sample statistic is too extreme to be considered plausible under the assumption that our estimated value equals the true population value.

The level of significance (\(\alpha\)) is found by subtracting the confidence level from one. For a 98% confidence interval, \( \alpha = 1 - \frac{98}{100} = 0.02 \). To get the critical value, you would then divide \(\alpha\) by two (for the two-tailed test) and look up this \(\alpha/2\) value in a z-table or use statistical software to find the z-score that corresponds to it. The resulting z-score is what we call the critical value. In this instance, the critical value for a 98% confidence interval is approximately 2.33. This value helps to set the range on either side of the sample mean for which we can say we're 98% confident the true population mean lies within.

Sample Standard Deviation

The sample standard deviation is a measure of the amount of variability or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range.

In the context of constructing a confidence interval, the sample standard deviation is used to estimate the variability inherent in the sample. It's the square root of the sum of the squared deviations from the mean divided by the number of observations minus one (the sample size minus one). This sample statistic is a critical component as it directly influences the standard error. If the standard deviation is large, indicating that the data is spread out, our standard error will also be larger, implying less precision in our estimate of the population mean. In the example given, a sample standard deviation of \(\(20\) suggests a moderate spread of individual expenses on graduation gifts around the sample mean of \(\)55.05\).

Sample Size

Sample size is the number of observations in a sample and it plays a vital role in statistics. It is denoted by \(n\). A larger sample size generally leads to more precise estimates of population parameters, like the mean or standard deviation, because it tends to reduce the standard error. You can think of it as getting more opinions before you make a decision; the more information you have, the better your estimation will be.

In the formula for standard error, the sample size appears in the denominator under the square root. This means as the sample size increases, the standard error decreases. It's a relationship that indicates the preciseness of your sample mean as an estimate of the population mean increases with more data points. For instance, with a large sample size of 2815, as in the example exercise, the impact of an individual value on the overall mean is minimized, thus providing a more stable and reliable estimate of the population parameter.