91Ó°ÊÓ

A laptop computer maker uses battery packs supplied by two companies, Aand B. While both brands have the same average battery life between charges (LBC), the computer maker seems to receive more complaints about shorter \(\mathrm{LBC}\) than expected for battery packs supplied by company \(\mathrm{B}\). The computer maker suspects that this could be caused by higher variance in \(\mathrm{LBC}\) for Brand \(B\). To check that, ten new battery packs from each brand are selected, installed on the same models of laptops, and the laptops are allowed to run until the battery packs are completely discharged. The following are the observed LBCs in hours. Test, at the \(5 \%\) level of significance, whether the data provide sufficient evidence to conclude that the LBCs of Brand \(B\) have a larger variance that those of Brand \(A\).

Step by step solution

01

State the Hypotheses

We need to check if Company B's battery packs have a higher variance in LBC than Company A's. Formally, this can be stated as:- Null Hypothesis \( H_0 \): \( \sigma_A^2 = \sigma_B^2 \)- Alternative Hypothesis \( H_1 \): \( \sigma_A^2 < \sigma_B^2 \)This is a one-tailed test for variance comparison.

02

Calculate Sample Variances

Suppose the LBCs for Brand A are \([t_1, t_2, ..., t_{10}]\) and for Brand B are \([s_1, s_2, ..., s_{10}]\). First, calculate the sample variance for each brand using the formula:\[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n}(x_i - \bar{x})^2 \]Calculate the variance for both brands using their respective LBCs.

03

Compute the Test Statistic

The test statistic for comparing two variances is the F-ratio, given by:\[ F = \frac{s_B^2}{s_A^2} \]where \( s_B^2 \) and \( s_A^2 \) are the sample variances for Brand B and Brand A, respectively. Compute the F-value using the variances calculated in Step 2.

04

Determine the Critical Value

For a one-tailed F-test at the 5% significance level with \( n_A - 1 \) and \( n_B - 1 \) degrees of freedom, find the critical value \( F_{\alpha} \) from an F-distribution table. For \( n_A = n_B = 10 \), degrees of freedom are 9 and 9.

05

Make the Decision

Compare the calculated F-value with the critical value:- If \( F > F_{\alpha} \), reject the null hypothesis \( H_0 \).- Otherwise, fail to reject \( H_0 \).This will tell us if there is sufficient evidence to conclude that the variance of LBCs for Brand B is larger than that for Brand A.

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Key Concepts

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

Variance Comparison

Variance comparison is the process of evaluating the spread, or dispersion, of data points in different datasets. In our exercise, we have two brands of battery packs, Brand A and Brand B. The goal is to determine which brand has greater variability in their battery life between charges (LBC). Variance is a common measure of dispersion that tells us how much the data points differ from the mean. It is calculated as the average of the squared differences from the mean.
To make a variance comparison, we compute the sample variances for both Brand A and Brand B using the formula:

• Calculate the mean of the data set.
• Subtract the mean from each data point, then square the result.
• Sum all of these squared differences.
• Divide by the number of data points minus one (i.e., use the formula for sample variance).

Variance comparison helps identify if there is a higher level of inconsistency in one dataset over another. This is crucial when consistency and reliability are needed, such as in battery performance.

F-test

The F-test is a statistical method used to compare two variances to determine if they are significantly different. It's used here to find out if the variance of battery life for Brand B is larger than that for Brand A. The F-test involves calculating the F-ratio, which is the ratio of two sample variances. The formula for the F-ratio is:\[ F = \frac{s_B^2}{s_A^2} \]where \( s_B^2 \) and \( s_A^2 \) are the sample variances of Brand B and Brand A, respectively.
The F-test is particularly useful because it can be applied even when the sample sizes are small, making it ideal for our situation with only ten samples from each brand. To perform an F-test:

• Calculate the sample variances for each group.
• Divide these variances to get the F-value.
• Compare the F-value to the critical value from an F-distribution table.

This will tell us whether there is statistical evidence to reject the null hypothesis.

Statistical Significance

Statistical significance determines if the observed differences in data sets (here, variance of battery life) are due to chance or if there is a genuine difference between groups. In the context of our exercise, it helps determine if there is substantial evidence to conclude that Brand B has a higher variance than Brand A.
The significance level, often denoted by \( \alpha \), is a threshold chosen by the researcher. Commonly set at 5% (0.05), it represents the probability of rejecting the null hypothesis when it is actually true (a Type I error).
The decision rule for statistical significance is straightforward:

• If our computed F-value is greater than the critical value from the F-distribution table, we conclude that the test result is statistically significant.
• This implies that there is sufficient evidence that the variance of Brand B's LBCs is greater than that of Brand A.
• If it is not significant, we cannot make such a claim.

Understanding statistical significance helps ensure our conclusion is reliable and supports informed decision-making.

Null Hypothesis

The null hypothesis is a fundamental concept in hypothesis testing. It represents a statement of no effect or no difference, serving as a baseline or default position that there is no relationship between variables. In our context, the null hypothesis \( H_0 \) is that the variances of the LBCs for both Brand A and Brand B are equal: \( \sigma_A^2 = \sigma_B^2 \).
Testing the null hypothesis involves using statistical tests, like the F-test, to determine if there is enough evidence to reject it. Here's how the process works:

• We calculate our test statistic (the F-value) based on our samples.
• We compare this statistic to a critical value.
• If the F-value is greater, it suggests the null may not hold, indicating a possible true difference in variances.

The null hypothesis is never proven true; it can only be rejected or not rejected. This concept is crucial as it guides decision-making in statistical testing, helping to avoid incorrect conclusions.