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Chapter 10: Problem 5

The average expenditure on Valentine's Day was expected to be \(\$ 100.89\) (USA Today. February 13,2006 ). Do male and female consumers differ in the amounts they spend? The average expenditure in a sample survey of 40 male consumers was \(\$ 135.67,\) and the average expenditure in a sample survey of 30 female consumers was \(\$ 68.64 .\) Based on past surveys, the standard deviation for male consumers is assumed to be \(\$ 35,\) and the standard deviation for female consumers is assumed to be \(\$ 20\) a. What is the point estimate of the difference between the population mean expenditure for males and the population mean expenditure for females? b. \(\quad\) At \(99 \%\) confidence, what is the margin of error? c. Develop a \(99 \%\) confidence interval for the difference between the two population means.

Short Answer

Expert verified
Point estimate is 67.03; margin of error is 17.068; confidence interval is (49.962, 84.098).

Step by step solution

01

Understand the Problem

We need to find the statistical evidence on whether male and female consumers spend differently on Valentine's Day. Specifically, we will calculate the point estimate of difference, the margin of error at 99% confidence, and then construct a confidence interval for the difference in their expenditures.
02

Calculate Point Estimate of the Difference

The point estimate of the difference between the population means is calculated using the difference between the sample means of males and females. Let \( \bar{x}_1 = 135.67 \) and \( \bar{x}_2 = 68.64 \). The point estimate \( \hat{\mu}_1 - \hat{\mu}_2 \) is given by \( \bar{x}_1 - \bar{x}_2 = 135.67 - 68.64 = 67.03 \).
03

Calculate the Standard Error of the Difference

The standard error \( SE \) for the difference between two population means is calculated using the formula: \[ SE = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} \] where \( \sigma_1 = 35 \), \( \sigma_2 = 20 \), \( n_1 = 40 \), \( n_2 = 30 \). Plugging in the values: \( SE = \sqrt{\frac{35^2}{40} + \frac{20^2}{30}} = \sqrt{30.625 + 13.333} = \sqrt{43.958} \approx 6.627 \).
04

Find the Critical Value for 99% Confidence

At 99% confidence, we use the z-distribution. The critical value \( z^* \) for 99% confidence is approximately 2.576.
05

Calculate Margin of Error

The margin of error (ME) is calculated as \( ME = z^* \times SE \). Substituting the values: \( ME = 2.576 \times 6.627 \approx 17.068 \).
06

Construct the Confidence Interval

Finally, the 99% confidence interval for the difference in population means is given by: \[ (\hat{\mu}_1 - \hat{\mu}_2) \pm ME \] \[ = 67.03 \pm 17.068 \] Thus, the confidence interval is \( (49.962, 84.098) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Point Estimate
The point estimate is a single value that serves as a best guess of an unknown population parameter. In the context of comparing expenditures between male and female consumers, the point estimate refers to the guessed difference in average spending between the two groups. To find this, we take the average spending for males and subtract the average spending for females.
  • Male average: \( \bar{x}_1 = 135.67 \)
  • Female average: \( \bar{x}_2 = 68.64 \)
  • Point estimate of the difference: \( 135.67 - 68.64 = 67.03 \)
This tells us that, based on the samples, male expenses were approximately $67.03 more than female expenses.
Standard Error
The standard error is a measure of the dispersion of sample statistics. In cases of comparing two means, it provides insight into how much the sample means are expected to vary. It's calculated from the standard deviations of both samples and their sizes.
  • Standard deviation for males: \( \sigma_1 = 35 \)
  • Standard deviation for females: \( \sigma_2 = 20 \)
  • Sample sizes: \( n_1 = 40 \,\text{and}\, n_2 = 30 \)
The formula to calculate the standard error of the difference between means is:\[SE = \sqrt{\frac{35^2}{40} + \frac{20^2}{30}} \approx 6.627\] A smaller standard error indicates more precise estimates of the population mean difference.
Margin of Error
The margin of error represents the range in which the true population parameter is likely to fall. It accounts for sampling variability and is computed using the standard error and the critical value.
  • Standard error: \( SE \approx 6.627 \)
  • Critical value for 99% confidence: \( z^* \approx 2.576 \)
The margin of error (ME) is calculated by:\[ME = z^* \times SE \approx 2.576 \times 6.627 \approx 17.068\] This means the estimated difference can reasonably differ by this amount from the true difference.
Critical Value
The critical value defines how many standard errors a parameter can be away from the sample mean and still be considered typical (i.e., within the confidence interval). It is derived from the z-distribution for large samples. For a 99% confidence level, the critical value is approximately 2.576. This means that in constructing a confidence interval, we account for the extreme case scenarios where the sample mean could diverge from the population mean by up to 2.576 standard errors. This parameter is crucial in determining how wide the confidence interval will be, impacting the precision of the estimate.
Population Mean Difference
The population mean difference is what we aim to estimate using sample data. It is an estimate of how much one group's mean differs from another鈥檚 in the entire population.In this exercise, male and female consumers exhibit different spending behaviors, and we're interested in the actual average spending difference for the whole population, not just the samples taken:
  • Point estimate of the difference: \( 67.03 \)
  • Using margin of error: \( ME = 17.068 \)
The 99% confidence interval for the population mean difference is:\[67.03 \pm 17.068 \]Thus, we assert that the true mean difference likely falls within \( (49.962, 84.098) \). This interval gives us a range for the actual difference between male and female spending although it's calculated from our limited sample.

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Most popular questions from this chapter

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