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 greater than 63% of those with diabetes who undergo this surgery will recover from diabetes? Use the a=0.10 level of significance and the P-value method.

Question

Hv0 : [?]
Hv1 : [?]

Left/right or two tailed test?

GPT-4o mini · Answer

To analyze whether greater than 63% of patients with Type 2 diabetes recover from diabetes after undergoing vertical banded gastroplasty, we need to set up the hypotheses.

Hypotheses:

Null Hypothesis (H0): The proportion of patients who recover from diabetes, $ p $, is equal to 0.63 (i.e., $ p = 0.63 $).
Alternative Hypothesis (H1): The proportion of patients who recover from diabetes, $ p $, is greater than 0.63 (i.e., $ p > 0.63 $).

Statement:

$ H_0: p = 0.63 $
$ H_1: p > 0.63 $

Type of Test:

Since we are interested in whether the recovery rate is greater than 63%, this is a one-tailed test (specifically, a right-tailed test).

Once we have the hypotheses and the type of test defined, we can calculate the test statistic and determine the p-value to see if we can reject the null hypothesis at the alpha level of 0.10.

Answer

(b) Compute the value of the test statistic. Round the answer to two decimal places.

GPT-4o mini · Answer

To compute the test statistic for this proportion test, we'll use the formula for the z-test statistic for a single proportion:

\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]

Where:

$\hat{p}$ = sample proportion of successes (those who recovered)
$p_0$ = hypothesized population proportion under the null hypothesis
$n$ = sample size

From the information provided:

The number of patients who recovered from diabetes, $x = 59$
The total number of patients studied, $n = 82$
The hypothesized proportion, $p_0 = 0.63$

First, we need to calculate the sample proportion $\hat{p}$:

\[ \hat{p} = \frac{x}{n} = \frac{59}{82} \approx 0.7183 \]

Now we can compute the z-test statistic:

Calculate the standard deviation for the null hypothesis proportion:

\[ \sqrt{\frac{p_0(1 - p_0)}{n}} = \sqrt{\frac{0.63 \times (1 - 0.63)}{82}} = \sqrt{\frac{0.63 \times 0.37}{82}} = \sqrt{\frac{0.2331}{82}} \approx \sqrt{0.002840} \approx 0.0533 \]

Compute the z value:

\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} = \frac{0.7183 - 0.63}{0.0533} \approx \frac{0.0883}{0.0533} \approx 1.60 \]

Conclusion:

The value of the test statistic $z$ is approximately 1.60 (rounded to two decimal places).

Answer

No z=1.68 use this to compute the P-Value. round to at least 4 decimal places

GPT-4o mini · Answer

To compute the p-value for the z-test statistic of $ z = 1.68 $, we can use a standard normal distribution table or a calculator to find the area to the right of the z-value because we are conducting a one-tailed test.

Finding the p-value: The p-value represents the probability of observing a test statistic as extreme as, or more extreme than, $ z = 1.68 $ under the null hypothesis $ H_0 $.
Using a standard normal distribution table or calculator:
- We can find the area to the left of $ z = 1.68 $ and subtract it from 1 to find the right-tail area.
- Looking up $ z = 1.68 $ in the Z-table gives us the value of approximately $ 0.9535 $ (the area to the left).
- To find the area to the right (the p-value):

\[ \text{p-value} = 1 - P(Z < 1.68) = 1 - 0.9535 \approx 0.0465 \]

Conclusion:

The p-value is approximately 0.0465 (rounded to four decimal places).

Answer

Do we reject Hv0?

GPT-4o mini · Answer

To determine whether to reject the null hypothesis ($ H_0 $), you need to compare the p-value to the significance level ($ \alpha $).

In this case:

The p-value we calculated is approximately 0.0465.
The significance level $ \alpha $ is given as 0.10.

Decision Rule:

If the p-value $ < \alpha $, we reject $ H_0 $.
If the p-value $ \geq \alpha $, we fail to reject $ H_0 $.

Conclusion:

Since $ 0.0465 < 0.10 $, we reject the null hypothesis $ H_0 $.

This provides statistically significant evidence to support the alternative hypothesis $ H_1 $: that greater than 63% of patients with Type 2 diabetes who undergo vertical banded gastroplasty will recover from diabetes.

Answer

e) State a conclusion. There [is or isnt?] enough evidence to conclude that the percentage of people with diabetes who undergo the surgery and recover from diabetes is greater than 63%.

GPT-4o mini · Answer

There **is** enough evidence to conclude that the percentage of people with diabetes who undergo the surgery and recover from diabetes is greater than 63%. This conclusion is based on the results of the hypothesis test, where we rejected the null hypothesis in favor of the alternative hypothesis at the $ \alpha = 0.10 $ significance level.