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Testing a headache remedy Studies that compare treatments for chronic medical conditions such as headaches can use the same subjects for each treatment. This type of study is commonly referred to as a crossover design. With a crossover design, each person crosses over from using one treatment to another during the study. One such study considered a drug (a pill called Sumatriptan) for treating migraine headaches in a convenience sample of children. The study observed each of 30 children at two times when he or she had a migraine headache. The child received the drug at one time and a placebo at the other time. The order of treatment was randomized and the study was double-blind. For each child, the response was whether the drug or the placebo provided better pain relief. Let \(p\) denote the proportion of children having better pain relief with the drug, in the population of children who suffer periodically from migraine headaches. Can you conclude that \(p>0.50,\) with more than half of the population getting better pain relief with the drug, or that \(p<0.50\), with less than half getting better pain relief with the drug (i.e., the placebo being better)? Of the 30 children, 22 had more pain relief with the drug and 8 had more pain relief with the placebo. a. For testing \(\mathrm{H}_{0}: p=0.50\) against \(\mathrm{H}_{a}: p \neq 0.50\), show that the test statistic \(z=2.56\). b. Show that the P-value is 0.01 . Interpret. c. Check the assumptions needed for this test, and discuss the limitations due to using a convenience sample rather than a random sample.

Step by step solution

01

Identify hypotheses

We are given two hypotheses: the null hypothesis, \(\mathrm{H}_{0}: p = 0.50\), and the alternative hypothesis, \(\mathrm{H}_{a}: p eq 0.50\). Here, \(p\) represents the proportion of all children who experience better pain relief from the drug than from the placebo.

02

Calculate the sample proportion

The proportion of children in the sample who experienced better pain relief with the drug is \(\hat{p} = \frac{22}{30} = 0.733\).

03

Determine standard error for the test statistic

The standard error \(SE\) is calculated using the formula for a proportion: \[ SE = \sqrt{\frac{p_0(1 - p_0)}{n}} \]where \(p_0 = 0.50\), the hypothesized population proportion, and \(n = 30\), the sample size.Substitute the values:\[ SE = \sqrt{\frac{0.50 \times 0.50}{30}} = \sqrt{\frac{0.25}{30}} = \sqrt{\frac{1}{120}} \approx 0.091\]

04

Compute the test statistic (z)

Now, calculate the test statistic using:\[ z = \frac{\hat{p} - p_0}{SE} \]Substitute the values:\[ z = \frac{0.733 - 0.50}{0.091} = \frac{0.233}{0.091} \approx 2.56\]

05

Find the P-value

Using the calculated z-score of 2.56, look up the corresponding P-value for a two-tailed test in the standard normal distribution table, which is approximately 0.01.

06

Interpret the P-value

Since the P-value of 0.01 is less than 0.05, the significance level, we reject the null hypothesis. This suggests that there is a statistically significant difference between the drug and placebo, indicating that the drug provides better pain relief.

07

Check assumptions and discuss limitations

For a valid z-test, the sample should be random. However, the study used a convenience sample, which may introduce bias affecting the generalizability of the results. Additionally, the large z-value and corresponding low P-value suggest that the sample size was sufficient to support the assumptions of normal approximation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Crossover Design

A crossover design is a robust way to study treatments using the same participants for both conditions. This method is quite efficient because each participant serves as their own control. Since participants "crossover" to receive all treatments under study, it minimizes variability that can occur from differences between individual participants.

In the headache remedy study, each child was observed twice: once when they received the drug and once with a placebo. The double-blind aspect ensured that neither the participants nor the researchers knew which treatment was being administered at any time, reducing potential bias.

Having participants experience both treatments and in random order helps balance out any carry-over effects where the first treatment might affect the outcome of the second. Despite these strengths, this design can be limited by its reliance on a convenience sample, which may not represent the wider population accurately.

P-value Interpretation

The P-value is a critical concept in hypothesis testing for evaluating whether the observed data disagree with the null hypothesis. In our study evaluating headache relief, the P-value is calculated as 0.01.

This P-value reflects the probability of observing such extreme results, or more, assuming the null hypothesis (that the drug has the same effect as the placebo) is true. A P-value less than 0.05, as in this instance, leads us to reject the null hypothesis with a certain level of confidence.

In plain terms, a P-value of 0.01 suggests a very low likelihood of these results occurring by chance alone. This supports the view that there truly is a substantial difference between the pain relief offered by the drug versus the placebo, favoring the alternative hypothesis.

Standard Error Calculation

The standard error (SE) quantifies the variation or error expected when estimating a population parameter based on a sample. Calculating the standard error is crucial for determining the confidence we can place in our sample's representation of the population.

For the headache remedy study, the formula used was:\[SE = \sqrt{\frac{p_0(1 - p_0)}{n}} \]where \( p_0 = 0.50 \) and \( n = 30 \).
This resulted in an SE of approximately 0.091.

This calculation helps us gauge how much the sample proportion might differ from the true population proportion due to sampling variability. By understanding this variability, we were able to compute a test statistic that assists in hypothesis testing, allowing us to determine the significance of our study results. This step in our analysis is pivotal in validating the findings from the sample.