91Ó°ÊÓ

An article in the November 1983 Consumer Reports compared various types of batteries. The average lifetimes of Duracell Alkaline AA batheries and Eveready Energizer Alkaline AA batteries were given as \(4.1\) hours and \(4.5\) hours, respectively. Suppose these are the population average lifetimes. a. Let \(\bar{X}\) be the sample average lifetime of 100 Duracell batteries and \(\bar{Y}\) be the sample average lifetime of 100 Eveready batteries. What is the mean value of \(\bar{X}-\bar{Y}\) (i.e., where is the distribution of \(\bar{X}-\bar{Y}\) centered)? Horv does your answer depend on the specitied sample sizes? h. Suppose the population standard deviations of lifetime are \(1.8\) hours for Duracell batteries and \(2.0\) hours for Eveready batteries. With the sample sizes given in part (a), what is the variance of the statistic \(\bar{X}-\bar{Y}\), and what is its standard deviation? c. For the sample sizes given in part (a), draw a picture of the approximate distribution curve of \(\bar{X}-\bar{Y}\) (include a measurement scale on the horizontal axis). Would the shape of the curve necessarily be the same for sample sizes of 10 batteries of each type? Explain.

Step by step solution

01

Understanding Mean Difference

To find the mean value of \(\bar{X} - \bar{Y}\), where \(\bar{X}\) is the sample average of Duracell batteries and \(\bar{Y}\) is the sample average of Eveready batteries, we use the population means directly. The mean difference \(\mu_{\bar{X}} - \mu_{\bar{Y}} = 4.1 - 4.5 = -0.4\). This value does not depend on the specified sample sizes; mean differences for population values remain constant regardless of sample sizes.

02

Calculating Variance of Mean Difference

The variance of \(\bar{X} - \bar{Y}\) is given by the formula: \(\text{Var}(\bar{X} - \bar{Y}) = \frac{\sigma_X^2}{n} + \frac{\sigma_Y^2}{m}\), where \(\sigma_X\) and \(\sigma_Y\) are the population standard deviations, and \(n\) and \(m\) are the sample sizes for Duracell and Eveready, respectively. Substitute \(\sigma_X = 1.8\), \(\sigma_Y = 2.0\), \(n = 100\), and \(m = 100\), we get \(\text{Var}(\bar{X} - \bar{Y}) = \frac{1.8^2}{100} + \frac{2.0^2}{100} = 0.324 + 0.4 = 0.724\).

03

Calculating Standard Deviation of Mean Difference

The standard deviation of the mean difference is the square root of the variance. Therefore, \(\text{SD}(\bar{X} - \bar{Y}) = \sqrt{0.724} \approx 0.8502\).

04

Describing Distribution Curve

The distribution of \(\bar{X} - \bar{Y}\) is approximately normal due to the Central Limit Theorem, given the large sample size. On a horizontal measurement scale, it is centered at \(-0.4\) with a standard deviation of approximately \(0.8502\). For smaller sample sizes, such as 10, the curve may be less normal, and the variance of the estimates would increase, altering the reliability and precision of the distribution characteristics.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Mean Difference

When comparing two sets of data, such as the average lifetimes of Duracell and Eveready batteries, the mean difference tells us how much better or worse one set is compared to the other. In this context, the mean difference is calculated by subtracting the population mean of Eveready batteries from that of Duracell batteries. This gives us a result of

• \(ar{X} - \bar{Y} = 4.1 - 4.5 = -0.4\) hours.

This negative result means that, on average, Duracell batteries last 0.4 hours less than Eveready batteries. What's important to remember is that this mean difference remains the same, irrespective of the sample sizes. This is because population means will not change when you're looking at sample averages instead of individual data points.
Understanding mean difference is crucial when making decisions based on population averages, as it provides a quantifiable comparison between two datasets. It directly informs how expectations might differ between groups, and gives insight into whether observed differences are significant.

Variance Calculation

Variance is a measure of how much the data points in a set differ from the mean. When examining the mean difference between two samples, knowing the variance helps us understand the spread of the data and how much the average differs between the two groups. The formula for the variance of the difference in sample means is:

• \(\text{Var}(\bar{X} - \bar{Y}) = \frac{\sigma_X^2}{n} + \frac{\sigma_Y^2}{m}\)

Here, \(\sigma_X\) and \(\sigma_Y\) are the population standard deviations for Duracell and Eveready batteries, respectively, and \(n\) and \(m\) are the sample sizes. Substituting these values (\(\sigma_X = 1.8\), \(\sigma_Y = 2.0\), \(n = 100\), and \(m = 100\)), we calculate:

• \(\text{Var}(\bar{X} - \bar{Y}) = \frac{1.8^2}{100} + \frac{2.0^2}{100} = 0.324 + 0.4 = 0.724\)

This variance indicates how spread out the average differences are likely to be within the samples studied. Larger variances suggest that there is more variability in lifetime durations, which may affect the reliability of the comparison.

Central Limit Theorem

The Central Limit Theorem (CLT) is a powerful principle in statistics. It's what allows us to assume that the distribution of sample means will be approximately normal, even if the population distribution isn’t. For large sample sizes, such as the 100 batteries from each brand in this exercise, the CLT assures that the distribution of the difference \((\bar{X}-\bar{Y})\) will tend to be normal. This assumption simplifies analysis and inference, providing a framework with predictable behavior.
The key point is that the distribution of \((\bar{X}-\bar{Y})\) is centered around the calculated mean difference with a variance of \(0.724\). This normal distribution characteristic helps visualize the outcomes, and predicts where sample means will likely fall, hence supporting better decision-making. However, for smaller sample sizes, such as 10 batteries, the approximation to the normal distribution may not be as accurate, and thus it might yield less precise estimates.

Standard Deviation

The standard deviation is essentially the square root of the variance and provides a measure of dispersion in the data relative to the mean. In the case of this exercise, we're looking at the standard deviation of the mean difference \((\bar{X} - \bar{Y})\), which tells us how much the average lifetime of these batteries might naturally fluctuate.
Calculating the standard deviation from the variance \(0.724\), we have:

• \(\text{SD}(\bar{X} - \bar{Y}) = \sqrt{0.724} \approx 0.8502\)

This process reveals the average extent to which the battery lifetimes differ from the mean difference. With a standard deviation of approximately \(0.8502\) hours, we can infer that when picking a random sample, the difference in mean lifetimes is expected to vary by this amount. Understanding the standard deviation is critical for gauging consistency and reliability in the observed differences between the two battery types.