A cell phone company offers two plans to it subscribers. At...

A cell phone company offers two plans to it subscribers. At the time new subscribers sign up, they are asked to provide some demographic information. The mean yearly income for a sample of 40 subscribers to Plan $\mathrm{A}$ is $\$ 57,000$ with a standard deviation of $\$ 9,200 .$ This distribution is positively skewed; the actual coefficient of skewness is 2.11. For a sample of 30 subscribers to Plan B the mean income is $\$ 61,000$ with a standard deviation of $\$ 7,100 .$ The distribution of Plan B subscribers is also positively skewed, but not as severely. The coefficient of skewness is 1.54 . At the .05 significance level, is it reasonable to conclude the mean income of those selecting Plan $\mathrm{B}$ is larger? What is the $p$ -value? Do the coefficients of skewness affect the results of the hypothesis test? Why?

Key Concepts

Null and Alternative Hypotheses

This concept involves formulating what is being tested. In testing whether the mean income for Plan B subscribers is larger, the null hypothesis typically states that there is no difference (or that Plan B's mean does not exceed Plan A's mean), and the alternative hypothesis states that Plan B’s mean income is higher. Clear hypotheses are essential to guide the subsequent test procedure.

p-value

The p-value represents the probability of obtaining test results at least as extreme as the observed outcome under the assumption that the null hypothesis is true. It provides a quantitative measure to decide whether the observed difference is statistically significant when compared to the chosen significance level.

t-test for Independent Samples

This statistical test compares the means from two independent groups to determine if there is a significant difference between them. It is especially useful when the sample sizes are relatively small, though it assumes that the distribution of sample means approximates normality, an assumption that is supported by the Central Limit Theorem for sufficiently large samples.

Central Limit Theorem

The Central Limit Theorem states that, given a sufficiently large sample size, the distribution of the sample mean will be approximately normally distributed, regardless of the population's distribution shape. This underpins the use of parametric tests like the t-test even when the population data exhibits skewness.

Coefficient of Skewness and Its Impact

The coefficient of skewness measures the asymmetry of a distribution. While high skewness can indicate that the distribution deviates from normality, the impact on hypothesis testing is often mitigated by the Central Limit Theorem when sample sizes are moderate or large. Thus, in many cases, the t-test remains robust even in the presence of skewed data.

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