Question

Olestra is a fat substitute approved by the FDA for use in snack foods. To investigate reports of gastrointestinal problems associated with olestra consumption, an experiment was carried out to compare olestra potato chips with regular potato chips ("Gastrointestinal Symptoms Following Consumption of Olestra or Regular Triglyceride Potato Chips," Journal of the American Medical Association \([1998]: 150-152\) ). Subjects were assigned at random to either the olestra chip group or the regular chip group. Of the 529 individuals in the regular chip group, 93 experienced gastrointestinal symptoms, whereas 90 of the 563 individuals in the olestra chip group experienced symptoms. Carry out a hypothesis test at the \(5 \%\) significance level to decide whether the proportion of individuals who experience symptoms after consuming olestra chips differs from the proportion who experience symptoms after consuming regular chips.

Short answer

Depending on the calculated Z score, and its comparison to the critical value, we decide whether or not to reject the null hypothesis. This would determine if the proportion of individuals who experience symptoms after consuming olestra chips significantly differs from the proportion who experience symptoms after consuming regular chips.

Step-by-step solution

Define the hypotheses

To start with, let's define our null hypothesis \(H_0\) and alternative hypothesis \(H_a\). The null hypothesis is that there is no difference in the proportion of individuals who experience symptoms after consuming olestra chips and regular chips. The alternative hypothesis is that there IS a difference. In formulas: \[H_0: p_{olestra} = p_{regular}\] \[H_a: p_{olestra} 𝑛𝑒𝑞 p_{regular}\]

Calculate pooled sample proportion

As the population standard deviation is not known, we calculate the pooled sample proportion (p) and standard error. Pooled sample proportion, \( p \), equals the total number of successes divided by the total number of observations: \[ p = \frac{x_1 + x_2}{n_1 + n_2} = \frac{93 + 90}{529 + 563} \]

Calculate the standard error

Once we have calculated the pooled sample proportion, we also need to find the standard error: \[ SE = \sqrt{ p * ( 1 - p ) * [ \frac{1}{n_1} + \frac{1}{n_2} ] } \] where \( n_1 \) and \( n_2 \) are the sizes of the two samples.

Calculate the z-score

Subtract the proportions, and divide by the standard error to find the z-score. The formula for this test statistic is: \[ Z = \frac{p_1 - p_2}{SE} \] where \(p_1\) and \(p_2\) are the observed sample proportions.

Determine the critical value

With a significance level of 5%, the critical value for a two-tailed test from the standard normal table is approximately ±1.96.

Make a decision

Compare the test statistic to the critical value, and reject the null hypothesis if our test statistic is more extreme than the critical value. Depending on the result, we will accept or reject the null hypothesis.

Key concepts

Pooled Sample Proportion

In hypothesis testing, particularly when comparing two proportions, we often use the pooled sample proportion. This concept is vital when the null hypothesis assumes that the two population proportions are equal. When we look at our olestra potato chips study, we're comparing the proportion of individuals who experienced symptoms after eating olestra chips versus regular chips. To calculate the pooled sample proportion, we combine the successes from both groups and divide by the total number of observations. This gives us a combined estimate that represents the overall proportion if the null hypothesis were true.

Here's the formula we use for the calculation: \[ p = \frac{x_1 + x_2}{n_1 + n_2} \]In the provided scenario, where \(x_1\) and \(x_2\) are the number of people with symptoms in the regular and olestra chip groups respectively, and \(n_1\) and \(n_2\) are the total number of individuals in each group, the pooled sample proportion is an essential step in determining the test statistic for our hypothesis test.

Standard Error Calculation

Once we have our pooled sample proportion, the next step is to calculate the standard error (SE) of the sampling distribution. Standard error is a measure of the variability or precision of the sample statistic. In the context of our chips study, it helps us understand how much the sample proportions might fluctuate from the true population proportion.

Here is the formula for standard error in a two-proportion hypothesis test: \[ SE = \sqrt{ p * (1 - p) * \left( \frac{1}{n_1} + \frac{1}{n_2} \right) } \]In this formula, \(p\) is the pooled sample proportion we calculated earlier, and \(n_1\) and \(n_2\) are the sizes of the two samples from the regular and olestra chip groups. It's important to understand that the standard error reflects how much we can expect the sample statistic to vary due to sampling variability, assuming the null hypothesis is true. Understanding standard error is central to interpreting the z-score which we calculate in the subsequent step of our hypothesis test.

Z-Score

Finally, we reach the crux of the hypothesis testing method—calculating the z-score. A z-score tells us how many standard errors a data point is from the mean. In hypothesis testing, it's used as a test statistic to determine whether or not to reject the null hypothesis.

The formula for the z-score in the context of comparing two sample proportions is: \[ Z = \frac{p_1 - p_2}{SE} \]Here, \(p_1\) and \(p_2\) are the sample proportions from the regular and olestra chip groups, while SE is the standard error we calculated earlier. This z-score is then compared to a critical value, which corresponds to our desired level of significance (5% in this case). If the absolute value of our calculated z-score is greater than the critical value (which is approximately ±1.96 for a two-tailed test at a 5% significance level), we would reject the null hypothesis. In the olestra chips study, this z-score calculation is key to concluding whether the two types of chips differ in the proportion of gastrointestinal symptoms reported by the consumers.

Ultimately, understanding how to calculate and interpret the z-score allows us to make informed decisions regarding our hypotheses. In educational terms, it's the bridge between theoretical probability and practical conclusion.