91Ó°ÊÓ

Sample proportions are a fundamental part of hypothesis testing. A sample proportion is simply the fraction or percentage of subjects in a sample that possess a certain characteristic. In the given problem, we are examining the proportion of individuals who get less than 6 hours of sleep in two different groups: truck drivers and a control group of non-transportation workers. To find the sample proportion for truck drivers, we calculate it as follows:

• Number of truck drivers with less than 6 hours of sleep: 35
• Total number of truck drivers surveyed: 203

The sample proportion for truck drivers is calculated as: \[ p_{\text{truck}} = \frac{35}{203} \approx 0.172 \]Similarly, for the control group, the sample proportion is: \[ p_{\text{control}} = \frac{35}{292} \approx 0.120 \]These simple calculations form the basis of comparing proportions between two groups.

The standard error (SE) is a measure of the variability or spread of sample proportion estimates from different samples. It's crucial in hypothesis testing because it tells us how much variability we can expect in the sample proportions if we repeated our survey multiple times. In our transportation worker example, calculating the SE for the difference in proportions helps gauge how much the estimated proportions might differ purely due to chance.
To calculate the standard error for the difference between two sample proportions, use:\[ SE = \sqrt{\frac{p_{\text{combined}} (1 - p_{\text{combined}})}{n_{\text{truck}}} + \frac{p_{\text{combined}} (1 - p_{\text{combined}})}{n_{\text{control}}}} \]Here, \( p_{\text{combined}} \) is the combined proportion:\[ p_{\text{combined}} = \frac{35+35}{203+292} = \frac{70}{495} \approx 0.141 \]Detecting changes in this combined proportion is essential for evaluating the results. Using this, we find:\[ SE \approx \sqrt{\frac{0.141 \times 0.859}{203} + \frac{0.141 \times 0.859}{292}} \approx 0.028 \]This tells us how much the proportion estimates are expected to vary due to random sampling.

The Z-score is a statistical measure that shows how far away a data point is from the mean of a population in standard deviation units. It's used in hypothesis testing to determine whether the difference between sample proportions is significant. Here, we want to know if the observed difference in the proportion of sleep-deprived individuals between truck drivers and the control group is statistically significant.
To calculate the Z-score, we use:\[ Z = \frac{(p_{\text{truck}} - p_{\text{control}})}{SE} \]Substituting the relevant values, we find:\[ Z = \frac{0.172 - 0.120}{0.028} \approx 1.86 \]The Z-score of 1.86 helps us compare the observed difference against random variation. This score will then be used to make a decision about our hypotheses.

The significance level, often denoted by \( \alpha \), is a threshold set by the researcher used to judge the statistical evidence against the null hypothesis. A common significance level is 0.05, meaning there's a 5% risk of concluding that a difference exists when there is none. In hypothesis testing, once we calculate the Z-score, we compare it to a critical value derived from the significance level.
For a two-tailed test with \( \alpha = 0.05 \), the critical Z-value is approximately \( \pm 1.96 \). The computed Z-score of 1.86 is less than the critical value of 1.96. This means the observed difference in proportions isn't large enough to warrant rejecting the null hypothesis.

• If the Z-score is greater than 1.96 or less than -1.96, we reject the null hypothesis.
• If it falls between -1.96 and 1.96, we fail to reject it.

In this case, since the Z-score does not exceed the critical value, we conclude there is not enough evidence to claim a significant difference in sleep deprivation between the two groups.