Question

The article "Caffeine Knowledge, Attitudes, and Consumption in Adult Women" (Journal of Nutrition Education [1992]: \(179-184\) ) reported the following summary statistics on daily caffeine consumption for a random sample of adult women: \(n=47, \bar{x}=215 \mathrm{mg}, s=\) \(235 \mathrm{mg}\), and the data values ranged from 5 to 1176 . a. Does it appear plausible that the population distribution of daily caffeine consumption is normal? Is it necessary to assume a normal population distribution to test hypotheses about the value of the population mean consumption? Explain your reasoning. b. Suppose that it had previously been believed that mean consumption was at most \(200 \mathrm{mg}\). Does the given information contradict this prior belief? Test the appropriate hypotheses at significance level. \(10 .\)

Short answer

In conclusion, plausibility of normal distribution for caffeine consumption cannot be definitively determined with limited information. And hypotheses testing reveals whether mean caffeine consumption statistically exceeds 200 mg. Given the input information, it's important to compute and analyse the test statistic correctly.

Step-by-step solution

Explore Data Distribution

First, let's evaluate whether it's plausible that the population distribution of daily caffeine consumption is normal. It is hard to determine only with given information because we have limited data. It is commonly checked by using the histogram, residuals, skewness, and kurtosis. In general, outliers could severely skew the distribution away from normal. Since we know that data range from 5 to 1176, indication of outliers is there. However, without additional data or graphs, we cannot definitively say whether the population distribution is normal or not. Furthermore, normality of the population distribution is not always crucial for testing hypotheses about the population mean if the sample size is significantly large. If the sample size is large (n > 30), the Central Limit Theorem states that the sample mean distribution will be approximately normal, regardless of the population distribution.

Establish Null and Alternative Hypotheses

Next, let's test whether the given information contradicts the prior belief that mean consumption was at most 200 mg. We do this by setting up our null and alternative hypotheses. The null hypothesis would be \(H_{0}\): \(\mu\) \(\leq\) 200 mg, meaning the mean caffeine consumption is at most 200 mg. The alternative hypothesis would be \(H_{1}\): \(\mu\) > 200 mg, meaning the mean caffeine consumption is more than 200 mg.

Conduct Hypothesis Testing

Perform the test of the hypothesis by calculating the test statistic, which is like a z-score, according to the formula: \(z\) = (\(\bar{x} - \mu_{0}\)) / (\(s / \sqrt{n}\)), where \(\bar{x}\) is the sample mean (215 mg), \(\mu_{0}\) is the population mean under the null hypothesis (200 mg), and \(s\) is the standard deviation (235 mg), \(n\) is the number of samples (47). Make sure to calculate the z-score correctly.

Check Significance Level

Compare the calculated test statistic with a critical value associated with the given significance level. If the test statistic is greater, we reject the null hypothesis, i.e., we have evidence to believe mean consumption is more than 200 mg. If the test statistic is less, we do not reject the null hypothesis, i.e., there's not enough evidence to refute the belief that the mean consumption is at most 200 mg.