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"Doctors Praise Device That. Aids Ailing Hearts" (Arsociated Press, November 9,2004\()\) is the headline of an article that describes the results of a study of the effectiveness of a fabric device that acts like a support stocking for a weak or damaged heart. In the study, 107 people who consented to treatment were assigned at random to either a standard treatment consisting of drugs or the experimental treatment that consisted of drugs plus surgery to install the stocking. After two years, \(38 \%\) of the 57 patients receiving the stocking had improved and \(27 \%\) of the patients receiving the standard treatment had improved. Do these data provide convincing evidence that the proportion of patients who improve is higher for the experimental treatment than for the standard treatment? 'Test the relevant hypotheses using a significance level of .05.

Step by step solution

01

Formulate the Hypotheses

Null hypothesis (H0): The proportion of patients who improve is the same for both treatments. H0: P1 = P2 \nAlternative hypothesis (H1): The proportion of patients who improve is higher for the experimental treatment than for the standard treatment. H1: P1 > P2

02

Find the Sample Proportions

Number of patients in experimental group (n1) = 57 (improved) out of 107 (total) \nProportion of patients who improved in experimental group (p1) = 38/57 = 0.67 \nNumber of patients in standard group (n2) = 50 (107-57) \nProportion of patients who improved in standard group (p2) = 27/50 = 0.54

03

Standard Error Calculation

Calculate the pooled proportion: \(p_p = (p1*n1 + p2*n2) / (n1 + n2)\) \nCalculate the standard error (SE) using the formula SE = sqrt(p_p * (1 - p_p) * [(1/n1) + (1/n2)])

04

Compute the Test Statistic

The Z score is given by: \(Z = (p1 - p2) / SE\)

05

Perform Hypothesis Test

Find the p-value corresponding to the calculated Z score. If the p-value is less than the level of significance (0.05), reject the null hypothesis.

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Key Concepts

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

Null Hypothesis

In hypothesis testing, the null hypothesis serves as the starting point. It is a statement that assumes no effect or no difference. That means, any observed effect in the data is due to random chance. In this particular exercise, the null hypothesis (H0) claims that there is no difference in the improvement rates between the standard treatment and the experimental one. Hence, the improvement proportions are equal: \(P_1 = P_2\). This hypothesis is what we initially assume to be true. In statistical tests, rejecting the null hypothesis implies that there is enough evidence to support the alternative hypothesis. It's like giving the existing condition the benefit of the doubt until there's sufficient evidence against it.

Alternative Hypothesis

The alternative hypothesis contradicts the null hypothesis. It suggests there is indeed an effect or difference. For this case, the alternative hypothesis (H1) asserts that the proportion of patients benefiting from the experimental treatment is greater than that of the standard treatment, or mathematically \(P_1 > P_2\). This one-sided alternative hypothesis is key when we believe there's a directional effect.

• It conveys what the researcher expects or wishes to prove.
• If the alternative hypothesis is true, it guides potential changes or interventions.

When conducting hypothesis testing, our goal is to see if we can gather enough data to reject the null hypothesis in favor of the alternative.

Significance Level

The significance level in a hypothesis test is a pivotal component. It helps us determine how much evidence we need to reject the null hypothesis confidently. It's usually denoted by \( \alpha \), such as 0.05, indicating a 5% risk of rejecting the null hypothesis when it's actually true (Type I error).
In this exercise, the significance level is set at 0.05. This means that if our p-value (probability of observing our sample data, given the null hypothesis) is less than 0.05, we will reject the null hypothesis.

• The choice of \( \alpha \) influences decision-making. Lower values increase caution against false rejections, while higher values aim for detecting true effects easier.
• Common significance levels are \(0.01\), \(0.05\), and \(0.10\).

Setting the significance level helps to balance between not missing a true effect and not raising false alarms.

Proportion Testing

Proportion testing is a statistical method used when we want to compare two different proportions. In this problem, it helps determine if the experimental treatment has a higher patient improvement rate than the standard treatment. Using proportion testing, we calculate sample proportions, pooled proportions, and standard error.

• Sample proportions identify the ratio of successes within each group, such as the 38% and 27% improvement rates here.
• A pooled proportion consolidates data from both groups together to figure out a collective improvement rate, assisting in error calculations.
• The standard error gives us an understanding of sampling variability, reflecting how much the sample proportion varies from the true population proportion.

These elements are part of computing a test statistic, like the Z score, to help make conclusions about the hypothesis. If the difference in proportions is statistically significant, it suggests an effect beyond random chance.