In a cover story, Bursiness Week published information about...

In a cover story, Bursiness Week published information about sleep habits of Americans (BusinessWeek, January 26,2004 ). The article noted that sleep deprivation causes a number of problems, including highway deachs. Fifty-one percent of adult drivers admit to driving while drowsy. A researcher hypothesized that this issue was an even bigger problem for night shift workers. a. Formulate the bypotheses that can be used to help determine whether more than 515 of the population of night shift workers admit to driving while drowsy. b. A sample of 400 night shift workers identified those who admitted to driving while drowsy. See the Drowsy file. Whar is the sample proportion? What is the $p$-value? c. At $a=.01$, what is your conclusion?

In a cover story, Bursiness Week published information about sleep habits of Americans (BusinessWeek, January 26,2004 ). The article noted that sleep deprivation causes a number of problems, including highway deachs. Fifty-one percent of adult drivers admit to driving while drowsy. A researcher hypothesized that this issue was an even bigger problem for night shift workers.
a. Formulate the bypotheses that can be used to help determine whether more than 515 of the population of night shift workers admit to driving while drowsy.
b. A sample of 400 night shift workers identified those who admitted to driving while drowsy. See the Drowsy file. Whar is the sample proportion? What is the $p$-value?
c. At $a=.01$, what is your conclusion?

Key Concepts

Significance Level and Decision Rule

The significance level (denoted by ?) is a threshold set by the researcher to judge whether the observed data are sufficiently unlikely under the null hypothesis. Common choices for ? include 0.05, 0.01, etc. If the p-value is less than or equal to the significance level, the null hypothesis is rejected in favor of the alternative hypothesis. Otherwise, there is insufficient evidence to reject the null hypothesis. This process helps control the probability of committing a Type I error, which is the error of falsely rejecting a true null hypothesis.

p-Value

The p-value is the probability, under the assumption that the null hypothesis is true, of obtaining a test statistic at least as extreme as the one observed. It provides a quantitative measure of the evidence against the null hypothesis. A smaller p-value indicates stronger evidence in favor of the alternative hypothesis. In hypothesis testing, the p-value is compared with the predetermined significance level to decide whether to reject the null hypothesis.

Hypothesis Formulation

This concept involves setting up a null hypothesis (H0) and an alternative hypothesis (Ha) before any data analysis is performed. In the context of testing proportions, the null hypothesis typically represents the status quo or a specified benchmark value (in this case, that the proportion is no greater than 51%), while the alternative hypothesis reflects the researcher's claim (that the proportion is greater than 51%). Proper formulation is crucial because it defines the framework for the statistical test and determines the direction of the test (one-sided or two-sided).

One-Sample Proportion Test

This is a statistical procedure used to determine whether the proportion observed in a sample significantly differs from a hypothesized population proportion. The test typically involves calculating the sample proportion from the observed data, then using a z-test (assuming the sample size is large enough for the normal approximation) to compare the observed proportion to the hypothesized value. The result is expressed as a z-score, which is then used to assess the significance of the difference.

Sample Proportion Calculation

The sample proportion is a statistic that represents the fraction of the sample that exhibits a particular characteristic, such as admitting to drowsy driving. It is calculated by dividing the number of affirmative responses by the total sample size. This value serves as an estimate of the true population proportion and is used as the basis for hypothesis testing in one-sample proportion tests.

Please give Ace some feedback