In 1955, Life Magazine reported that the 25 year-old mother...

In $1955,$ Life Magazine reported that the 25 year-old mother of three worked, on average, an 80 hour week. Recently, many groups have been studying whether or not the women's movement has, in fact, resulted in an increase in the average work week for women (combining employment and at-home work). Suppose a study was done to determine if the mean work week has increased. 81 women were surveyed with the following results. The sample mean was $83 ;$ the sample standard deviation was ten. Does it appear that the mean work week has increased for women at the $5 \%$ level?

Key Concepts

Null and Alternative Hypotheses

In hypothesis testing, the process starts by formulating the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis typically represents the status quo or an assumed value (in this case, that the mean work week is 80 hours), while the alternative hypothesis suggests that there is a change or difference (here, that the mean work week has increased above 80 hours). This formulation sets the stage for statistical testing.

Test Statistic

The test statistic is a standardized value that measures the discrepancy between the sample statistic (like the sample mean) and the hypothesized population parameter under the null hypothesis. It is calculated by taking the difference between the sample mean and the hypothesized mean and then dividing by the standard error. This value is used to determine how likely or unlikely the observed data is under the null hypothesis.

Standard Error

Standard error represents the estimated standard deviation of the sampling distribution of the sample mean. It is calculated by dividing the sample standard deviation by the square root of the sample size. The standard error quantifies the amount of variability associated with the estimate of the mean and is crucial in scaling the test statistic.

Significance Level and p-value

The significance level (commonly denoted as alpha and often set at 0.05) is the threshold for deciding whether the observed data is extreme enough to reject the null hypothesis. The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, given that the null hypothesis is true. If the p-value is lower than the significance level, it indicates strong evidence against the null hypothesis, leading to its rejection.

One-tailed Test

A one-tailed test is used when the research hypothesis predicts a specific direction of the effect, such as an increase in mean work hours. In this context, the test only considers the probability of observing a result in one direction (greater than the hypothesized mean), which often results in a more powerful test to detect an increase.

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