: Vertical banded gastroplasty is a surgical procedure that reduces the volume of the stomach in order to produce weight loss. In a recent study, 82 patients with Type 2 diabetes underwent this procedure, and 59  of them experienced a recovery from diabetes. Does this study provide convincing evidence that those with diabetes who undergo this surgery will recover from diabetes differs from 56% ? Use the a=0.01 level of significance with the P-value method and Excel.

To determine whether the study provides convincing evidence that the proportion of diabetes recovery after vertical banded gastroplasty differs from 56%, we will conduct a hypothesis test for a population proportion using the given data and the P-value method at a significance level of \(\alpha = 0.01\). We will use Excel to perform this test.

### Step-by-Step Calculation:

#### 1. State the Hypotheses:
- Null Hypothesis (\(H_0\)): \(p = 0.56\)
- Alternative Hypothesis (\(H_a\)): \(p \ne 0.56\)

This is a two-tailed test because we are checking if the recovery rate is different from 56%.

#### 2. Calculate the sample proportion (\(\hat{p}\)):
\[
\hat{p} = \frac{59}{82} \approx 0.7195
\]

#### 3. Calculate the test statistic (z):
The test statistic for a proportion is calculated using the formula:
\[
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}
\]
Where:
- \(\hat{p} = 0.7195\)
- \(p_0 = 0.56\)
- \(n = 82\)

Substitute these values into the formula:
\[
z = \frac{0.7195 - 0.56}{\sqrt{\frac{0.56 \cdot 0.44}{82}}} = \frac{0.1595}{\sqrt{0.2464 / 82}} = \frac{0.1595}{0.0546} \approx 2.92
\]

#### 4. Calculate the P-value:
Since this is a two-tailed test, we need to find the area under the standard normal curve for the z-value and then double it.

To calculate the P-value using Excel:
- Use the `NORM.S.DIST` function for the standard normal cumulative distribution.
- Calculate the P-value for the positive z-value.
- Double this value to account for both tails.

\[
\text{P-value} = 2 \times (1 - \text{NORM.S.DIST}(2.92, TRUE))
\]

In Excel:
\[
\text{P-value} = 2 \times (1 - \text{NORM.S.DIST}(2.92, TRUE)) \approx 2 \times (1 - 0.9982) = 2 \times 0.0018 = 0.0036
\]

#### 5. Make a Decision:
Compare the P-value to the significance level \(\alpha = 0.01\):
- If P-value \(\leq \alpha\), reject the null hypothesis.
- If P-value \(> \alpha\), fail to reject the null hypothesis.

In this case:
\[
\text{P-value} = 0.0036 < 0.01
\]

#### 6. Conclusion:
Since the P-value is less than the significance level (\(\alpha = 0.01\)), we reject the null hypothesis. 

### Conclusion:
There is convincing evidence at the \(\alpha = 0.01\) level of significance that the proportion of diabetes recovery after vertical banded gastroplasty differs from 56%.

Would you like further details or have any questions?

Here are five questions you might find interesting:
1. How does a two-tailed test differ from a one-tailed test in hypothesis testing?
2. Why is it important to determine the sample size before conducting a hypothesis test?
3. Can you explain how confidence intervals can be used to support the results of a hypothesis test?
4. What are the steps to perform a hypothesis test for a mean instead of a proportion?
5. How does the choice of the significance level (\(\alpha\)) impact the likelihood of committing Type I and Type II errors?

**Tip:** When performing hypothesis tests in Excel, remember to use the appropriate functions and ensure that you correctly interpret the results based on the context of your test.

Explore a hypothesis test for the proportion of diabetes recovery after vertical banded gastroplasty using Excel and the P-value method. Understand the calculation steps and decision-making process with a significance level of α = 0.01.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation