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When comparing two different types of automotive batteries, understanding how their lifespan differs on average is key. In this scenario, we aim to determine which battery, Type A or Type B, has a longer average life. By gathering sample data from twenty batteries of each type, we're able to calculate summary statistics like the mean ( X ext{ and }x_A=32.91 ext{ and } X_B=30.47 ext{. ) To evaluate which battery is superior regarding lifespan, examining the mean difference provides insights into performance. The initial calculation involves finding the sample mean difference, also known as D_x . This is done by subtracting the mean of Battery B from that of Battery A ( 32.91 - 30.47 = 2.44 ). Through this process, it becomes evident whether there's a significant advantage in choosing one battery over the other based on mean lifespan alone, making it crucial to proceed with further statistical analysis.

Pooled variance is a technique used to estimate a common variance for two or more groups. It’s particularly useful when you assume equal population variances, which is true in this exercise for Battery Types A and B.To calculate pooled variance, we first determine the individual variances of each battery group. The sample variance for Battery A is s_A^2 = (1.57)^2 = 2.4649, and for Battery B, s_B^2 = (1.74)^2 = 3.0276. By combining these, the pooled variance is calculated as (s^2_P). The formula is:\[s^2_P = rac{(s^2_A + s^2_B)}{2} = rac{(2.4649 + 3.0276)}{2} = 2.74625\]The resulting pooled variance captures both groups' data variability more robustly than using individual variances alone. It serves as a foundation for further steps like standard error and confidence intervals, providing a shared estimate of variance.

After determining the pooled variance, the standard error of the difference in means provides insight into the variability expected when comparing battery life spans. The standard error reflects how much the sample means (mean life of the batteries) would vary if we were to take multiple samples.To compute the standard error:1. Apply the pooled variance (s^2_P = 2.74625) calculated earlier.2. Use the formula for standard error, given by:\[s_{D_x} = \sqrt{\frac{s^2_P}{n_A} + \frac{s^2_P}{n_B}}\]Inserting the values yields:\[s_{D_x} = \sqrt{\frac{2.74625}{20} + \frac{2.74625}{20}} = \sqrt{0.1373125} = 0.370558\]This result, (0.370558), quantifies the uncertainty in our measurement of the average lifespan difference, guiding the construction of confidence intervals.

The t-distribution is essential when working with sample data, especially when the sample size is small and the population standard deviation is unknown. In comparing battery life, it helps form conclusions about means using sample data.Since we are interested in a 95% confidence interval for the mean difference between battery types, we use the t-distribution. Based on the degrees of freedom (df = 38)—the sum of sample sizes (20+20-2)—a t-score is consulted to find how values are distributed.For a 95% confidence level, this t-score is approximately 2.024. Subsequently, the confidence interval formula is:\[D_x \pm t \times s_{D_x}\]In this example, it translates to:\[2.44 \pm 2.024 \times 0.370558 = 2.44 \pm 0.749228\]This gives a confidence interval of (1.690772, 3.189228), meaning we are 95% sure the mean life of Battery A is between 1.690772 and 3.189228 units greater than Battery B. As the interval contains only positive numbers, Battery A is revealed to statistically have a longer mean lifespan.