91Ó°ÊÓ

An article in the November 1983 Consumer Reports compared various types of batteries. The average lifetimes of Duracell Alkaline AA batteries 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)? How does your answer depend on the specified sample sizes? b. 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

Calculate the Mean of XÌ„ and YÌ„

For part (a), the population average lifetime of Duracell batteries is given as \(4.1\) hours, and for Eveready batteries, it is \(4.5\) hours. The mean of a sample average \(\bar{X}\) is equal to the population mean \(\mu_X = 4.1\) hours. Similarly, the mean of \(\bar{Y}\) is \(\mu_Y = 4.5\) hours. Therefore, the mean of \(\bar{X} - \bar{Y}\) is \(\mu_X - \mu_Y = 4.1 - 4.5 = -0.4\). The answer does not depend on the sample size because the mean difference is based on population means.

02

Calculate the Variance of XÌ„ - YÌ„

For part (b), given that the population standard deviations are \(\sigma_X = 1.8\) hours for Duracell and \(\sigma_Y = 2.0\) hours for Eveready, and both sample sizes are 100. The variance of the difference \(\bar{X} - \bar{Y}\) is calculated as:\[ \text{Var}(\bar{X} - \bar{Y}) = \frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y} = \frac{1.8^2}{100} + \frac{2.0^2}{100} = \frac{3.24}{100} + \frac{4.0}{100} = 0.0324 + 0.040 = 0.0724. \]

03

Calculate the Standard Deviation of XÌ„ - YÌ„

Continuing with part (b), the standard deviation of \(\bar{X} - \bar{Y}\) is the square root of the variance:\[ \text{SD}(\bar{X} - \bar{Y}) = \sqrt{0.0724} \approx 0.269\] hours.

04

Describe the Distribution Curve of XÌ„ - YÌ„

For part (c), since both sample sizes are 100, by the Central Limit Theorem, \(\bar{X} - \bar{Y}\) is approximately normally distributed with mean \(-0.4\) and standard deviation \(0.269\). If the sample sizes were 10, the curve might be less normal and wider due to increased variability. The distribution will still center at \(-0.4\) but will be less symmetric, which might deviate from a perfect normal distribution, making the tails heavier.

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Key Concepts

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

Population Standard Deviation

Population standard deviation is a measure that captures the amount of variability or dispersion in a set of data points in a population. In the context of the battery exercise, it's used to understand the variation in lifetime hours of Duracell and Eveready batteries. When calculating statistics like variance or standard deviation for sample means, knowing the population standard deviation (symbolized as \( \sigma \)) helps in assessing how much the sample means might fluctuate from the true population mean.For example, Duracell's population standard deviation is given as \( 1.8 \) hours and Eveready’s as \( 2.0 \) hours. By understanding these values, we can make informed predictions about sample data behavior or sampling error.

Central Limit Theorem

The Central Limit Theorem (CLT) is fundamental in statistics. It profoundly influences how we approach sample distributions in practice. According to the CLT, when you have a large enough sample size, the distribution of the sample mean will approximate a normal distribution, regardless of the original population distribution. In the battery lifetime example, the theorem assures us that the difference in sample means, \( \bar{X} - \bar{Y} \), will be approximately normally distributed if our sample sizes are adequately large (e.g., 100 batteries each). This makes it easier to make predictions and draw conclusions about the population mean differences with reliability.

Variance Calculation

Calculating variance is crucial because it provides a numerical value for the dispersion within data sets. For sample means, understanding variance helps infer how closely a sample mean might reflect the population mean.For our exercise, we calculate the variance of \( \bar{X} - \bar{Y} \), the difference in sample means of two battery types. This is done using the formula:\[\text{Var}(\bar{X} - \bar{Y}) = \frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}\]where \( \sigma_X \) and \( \sigma_Y \) are the respective population standard deviations, and \( n_X \) and \( n_Y \) are the sample sizes. Understanding variance helps determine the consistency and reliability of data predictions.

Normal Distribution Approximation

Normal distribution approximation is a statistical method for describing data that clusters around a mean. Thanks to the Central Limit Theorem, even if population data is not normally distributed, the means of sufficiently large samples from that population will be normally distributed.In the discussed battery study, normal distribution approximation allows us to visualise the distribution of sample mean differences, \( \bar{X} - \bar{Y} \), with a graph resembling the bell curve. This approach is beneficial for examining how close our sample calculations are to what’s true for the entire population. If the sample sizes were a lot smaller (e.g., 10 batteries instead of 100), the approximation would be less accurate, resulting in a distribution that could be more skewed or "fatter" at the ends than a perfect normal distribution.