A company sends a random sample of 16 of its salespeople to...

A company sends a random sample of 16 of its salespeople to a course designed to increase their motivation and, hence, presumably their effectiveness. In the following year these people generated sales with an average value of $$\$ 625,000$$ and a sample standard deviation of $$\$ 80,000$$. During the same period, an independently chosen random sample of 10 salespeople who had not attended the course obtained sales with an average value of $$\$ 608,000$$ and a sample standard deviation of $$\$ 73,000$$. Assume that the two population distributions are normal and have the same variance. Find a $90 \%$ confidence interval estimate for the difference between the population mean sales for salespeople who attended the motivational course and for those salespeople who did not attend the course.

Key Concepts

Confidence Interval for the Difference of Means

A confidence interval for the difference between population means provides a range in which the true difference is likely to lie. It is computed by taking the observed difference in sample means and adding and subtracting a margin of error, which is determined by a critical t-value and the standard error of the difference.

Degrees of Freedom

Degrees of freedom represent the number of independent values in a calculation that are free to vary. In the context of a pooled two-sample t-test, the degrees of freedom are calculated as the sum of the sample sizes minus two, and they are critical for determining the correct shape of the t-distribution and the appropriate critical values.

t-Distribution

The t-distribution is a type of probability distribution that is symmetric and bell-shaped, similar to the normal distribution, but with heavier tails. It is especially useful in small sample scenarios where the population standard deviation is unknown, ensuring that the uncertainty introduced by estimating the variance is properly accounted for.

Two-Sample t-Test

The two-sample t-test is used to compare the means of two independent groups to determine if there is statistical evidence that the associated population means are significantly different. This technique underpins the analysis when assessing the effect of a treatment or intervention across two separate samples.

Pooled Variance Estimation

When the assumption of equal variances for the two populations is justified, the pooled variance method combines the sample variances into a single estimate of the common population variance. This approach increases the reliability of the variance estimate, particularly in studies involving small sample sizes.

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