Final answer:
To construct a 95% confidence interval of the population mean spending, the Central Limit Theorem is used to assume the distribution of the sample mean is approximately normal, allowing for the use of z-scores to calculate the critical value. The 95% confident that the true population mean spending on gifts during the entire holiday season falls between $522.61 and $577.39.
Step-by-step explanation:
To construct a 95% confidence interval of the population mean spending, we use the formula:
Confidence Interval = Sample Mean ± Margin of Error
The margin of error is calculated by multiplying the critical value (z-score) by the standard deviation of the sample, divided by the square root of the sample size.
In this case, since the sample size is larger than 30, the Central Limit Theorem can be used.
The Central Limit Theorem states that in large samples, the sampling distribution of the sample mean approaches a normal distribution, regardless of the shape of the population distribution.
This theorem allows us to assume that the distribution of the sample mean is approximately normal, and we can use z-scores to calculate the critical value.
Using the given information, the critical value for a 95% confidence level is approximately 1.96.
Therefore, the margin of error is:
Margin of Error = 1.96 * ($92 / sqrt(49))
≈ $27.39
Finally, we can construct the confidence interval:
Confidence Interval = $550 ± $27.39
= ($522.61, $577.39)
This means that we are 95% confident that the true population mean spending on gifts during the entire holiday season falls between $522.61 and $577.39.