Question

Independent random samples were selected from two quantitative populations, with sample sizes, means, and variances given in Exercises 13-14. Construct a 95\% upper confidence bound for \(\mu_{1}-\mu_{2}\). Can you conclude that one mean is larger than the other? If so, which mean is larger? $$\begin{array}{lcc}\hline & \multicolumn{2}{c} {\text { Population }} \\\\\cline { 2 - 3 } & 1 & 2 \\\\\hline \text { Sample Size } & 35 & 49 \\\\\text { Sample Mean } & 9.7 & 7.4 \\\\\text { Sample Variance } & 10.78 & 16.44 \\\\\hline\end{array}$$

Short answer

Answer: Yes, the mean of Population 1 is larger than the mean of Population 2.

Step-by-step solution

Find the point estimate and standard error of the difference in means

The point estimate for \(\mu_1 - \mu_2\) is given by the difference in sample means, \(\bar{X}_1 - \bar{X}_2\). The standard error for the difference in means can be found using the formula: $$SE = \sqrt{\frac{S_1^2}{n_1} + \frac{S_2^2}{n_2}}$$ where \(S_1^2\) and \(S_2^2\) are the sample variances, and \(n_1\) and \(n_2\) are the sample sizes. In this case, we have: \(n_1 = 35\), \(n_2 = 49\), \(\bar{X}_1 = 9.7\), \(\bar{X}_2 = 7.4\), \(S_1^2 = 10.78\), \(S_2^2 = 16.44\)

Calculate the point estimate and standard error

Now we plug in the values into the formulas mentioned above: Point Estimate: \(\bar{X}_1 - \bar{X}_2 = 9.7 - 7.4 = 2.3\) Standard Error: $$SE = \sqrt{\frac{10.78}{35} + \frac{16.44}{49}} = 0.846$$

Determine the critical value

Using a t-table for a 95\% confidence level, the critical value (t-score) can be found. Since we only need the upper bound, we will use a one-tailed t-test. For the degrees of freedom, we can use the conservative approach which is to use the smaller of \((n_1 - 1)\) and \((n_2 - 1)\): Degrees of freedom = \(min(n_1 - 1, n_2 - 1) = min(34, 48) = 34\) Now, we find the t-score for a one-tailed test with 95\% confidence and 34 degrees of freedom. From the t-table, we get: \(t=1.690\).

Calculate the confidence interval's upper bound

Now we can calculate the upper bound of the 95\% confidence interval with the following formula: Upper Bound = Point Estimate + (t-score × Standard Error) Upper Bound = \(2.3 + (1.690 × 0.846) = 4.129\)

Conclusion

The 95\% upper confidence bound of the difference between the two means, \(\mu_1 - \mu_2\), is 4.129. Since this value is positive and significantly above zero, we can conclude that \(\mu_1\) is greater than \(\mu_2\). Therefore, the mean of Population 1 is larger than the mean of Population 2.

Key concepts

Independent Random Samples

Understanding independent random samples is crucial when trying to infer population characteristics using sample data. such samples are collected in a way that the selection of a sample from one population does not influence the selection from another. This means the two sets of data don't affect each other and are not related. For instance, measuring the heights of students in two different classrooms would be considered independent because the measurement of one student's height in the first classroom doesn't affect the measurement in the second classroom.

In statistical analysis, ensuring that samples are independent is key to applying certain types of inferential tests, like the t-test for comparing means. If the samples weren't independent, we might need to use different methods, such as paired tests or repeated measures analysis, to account for the relationships between samples.

Upper Confidence Bound

The upper confidence bound is part of what statisticians call a confidence interval. A confidence interval gives an estimated range of values which is likely to include an unknown population parameter, the estimated range being calculated from a given set of sample data. When we say '95% upper confidence bound', we're talking about the upper limit of this range for a 95% confidence level.

This means that we are 95% confident that the true population mean difference between two independent random samples will not exceed this upper bound. It's important to note, however, that this doesn't imply there's a 95% chance that the true mean difference lies below this bound, but rather that the process of calculating the interval yields intervals that include the true mean difference 95% of the time over many repetitions.

T-Score

A t-score, also known as a t-statistic, is a type of standardized score used in statistics to determine the relative position of a sample mean to the hypothesized population mean, expressed in terms of standard error. It plays an essential role in hypothesis testing and confidence interval estimation when dealing with small sample sizes or unknown population variances.

Specifically, t-scores are used with t-distributions, which are symmetrical and bell-shaped but have fatter tails than the standard normal distribution. These fatter tails provide more conservative confidence intervals and hypothesis tests, which is necessary when making inferences from smaller samples. The t-score can tell us how far from the mean our observation is, in the context of standard errors.

Standard Error of the Difference in Means

The standard error of the difference in means measures the variability of the difference between two sample means. It's essential when we want to infer about the difference between two population means based on our samples. Calculating this standard error requires knowing both sample variances and sizes.

To interpret this, think of it as how much uncertainty there is around our estimate of the difference in population means. A larger standard error would indicate more variability and thus more uncertainty. On the other hand, a smaller standard error means less variability and more confidence in our point estimate of the difference. This standard error is a key component in computing confidence intervals for the difference in means, allowing us to estimate a range within which the true difference likely falls.