Chapter 9, Problem 56CR

To determine

Find the 98% confidence interval for the mean amount of money spent per graduation gift in 2007 and interpret it.

Answer to Problem 56CR

The 98% confidence interval for the mean amount of money spent per graduation gift in 2007 is (54.173, 55.927).

Explanation of Solution

Calculation:

The mean amount of money spent per graduation gift is $55.05 and the standard deviation is $20. The estimate was based on a sample of 2,815 consumers.

The 98% confidence interval for the population mean is calculated as follows:

The confidence interval for a population mean µ is appropriate in following cases:

• The sample size is large (generally, $n\ge 30$). That is, the population distribution is normal.
• The sample mean is $\overline{x}$, obtained from a simple random sample.
• The population standard deviation is unknown.

The general formula for finding the confidence interval for a population mean is, $\overline{x}\pm \left(t\text{ critical value}\right)\left(\frac{s}{\sqrt{n}}\right)$, where $\widehat{p}$ is the sample proportion, n is the sample size, and t critical value is based on  n –1 degrees of freedom.

Check conditions:

A sample of 2,815 consumers is considered. The sample size is large, and it is taken from a large population of consumers who reported that they bought one or more graduation gift in 2007. Thus, the population distribution is normal.

Here, the sample mean is $55.05 and standard deviation is $20 as obtained from the sample.

It is given that the sample is a representative of the population of Country A adults, who purchased graduation gift in 2007. Thus, it is reasonable to consider the sample as a random sample from the population.

The population standard deviation is unknown.

Thus, the confidence interval for a population mean µ is appropriate.

Software procedure:

Step-by-step procedure to obtain the confidence interval using MINITAB software:

• Choose Stat > Basic Statistics > 1-Sample t.
• In Summarized data, enter the Sample size as 2,815 and mean as 55.05.
• In Standard deviation, enter the value 20.
• In Options, enter Confidence level as 98.0.
• Choose not equal in alternative.
• Click OK in all dialogue boxes.

The output using the MINITAB software is given below:

From the MINITAB output, the confidence interval is (54.173, 55.927).

There is a 98% confidence that the mean amount of money spent per graduation gift in 2007 is lies between $54.173 and $55.927.