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In a recent study, 82 patients with type 2 diabetes underwent a procedure, and 59 recovered. Does this study provide convincing evidence that greater than 58% will recover? a=.10

What are the null and alternative hypothesis

5 answers

To determine if there is convincing evidence that greater than 58% of patients will recover, we can formulate the null and alternative hypotheses as follows:

Null Hypothesis (\(H_0\)): The proportion of patients who recover is less than or equal to 58%.
Mathematically, this can be stated as: \[ H_0: p \leq 0.58 \]

Alternative Hypothesis (\(H_a\)): The proportion of patients who recover is greater than 58%.
Mathematically, this can be stated as: \[ H_a: p > 0.58 \]

Given these hypotheses, you can perform a statistical test (such as a one-sample z-test for proportions) to analyze the data and draw conclusions regarding the population recovery rate based on your sample.

Compute the test statistics

To compute the test statistic for the hypothesis test regarding the proportion of patients who recover, we can use the formula for the z-test for proportions.

  1. Determine the sample proportion (\( \hat{p} \)): \[ \hat{p} = \frac{x}{n} \] where \( x \) is the number of successes (recoveries) and \( n \) is the total number of trials (patients).

    • In this case, \( x = 59 \) (patients who recovered) and \( n = 82 \) (total patients). \[ \hat{p} = \frac{59}{82} \approx 0.7195 \]

  2. Specify the claimed proportion in null hypothesis: \[ p_0 = 0.58 \]

  3. Calculate the standard error (SE) of the proportion: \[ SE = \sqrt{\frac{p_0(1 - p_0)}{n}} \] \[ SE = \sqrt{\frac{0.58(1 - 0.58)}{82}} = \sqrt{\frac{0.58 \times 0.42}{82}} \approx \sqrt{\frac{0.2436}{82}} \approx \sqrt{0.00297} \approx 0.0545 \]

  4. Calculate the z-test statistic: \[ z = \frac{\hat{p} - p_0}{SE} \] \[ z = \frac{0.7195 - 0.58}{0.0545} \approx \frac{0.1395}{0.0545} \approx 2.56 \]

Thus, the test statistic \( z \) is approximately 2.56.

You can now use this z-value to compare against the critical value or to find the p-value for your hypothesis test at the \( \alpha = 0.10 \) significance level.

Compute the p value

To compute the p-value for the test statistic \( z \approx 2.56 \), we can use standard normal distribution tables or statistical software/calculators.

Since we are conducting a one-tailed test (testing if the proportion is greater than 0.58), we want to find the area to the right of \( z = 2.56 \).

  1. Find the cumulative probability for \( z = 2.56 \): Using a standard normal distribution table or calculator, we find \( P(Z < 2.56) \).

    The cumulative probability for \( z = 2.56 \) is approximately: \[ P(Z < 2.56) \approx 0.9948 \]

  2. Calculate the p-value: The p-value is the area to the right of \( z = 2.56 \): \[ p\text{-value} = 1 - P(Z < 2.56) = 1 - 0.9948 = 0.0052 \]

Thus, the p-value is approximately 0.0052.

Interpretation:

Since the p-value (0.0052) is less than the significance level \( \alpha = 0.10 \), you would reject the null hypothesis. This provides convincing evidence that the proportion of patients who recover is greater than 58%.