Question

The body mass index (BMI) of an individual is a measure used to judge whether an individual is overweight or not. A BMI between 20 and 25 indicates a normal weight. In a survey of 750 men and 750 women, the Gallup organization found that 203 men and 270 women were normal weight. Is there a difference in the proportion of men and women who are normal weight? Use 0.01 as the level of significance.

What are we looking for and what are we going to do with this?

Answer 1

To determine if there is a difference in the proportion of men and women who are normal weight, we can use a two-proportion z-test. We compare the proportion of normal weight men to the proportion of normal weight women and analyze the test statistic to determine if the difference is statistically significant.

Step-by-step explanation:

To determine if there is a difference in the proportion of men and women who are normal weight, we can use a hypothesis test. We will compare the proportion of normal weight men to the proportion of normal weight women using a two-proportion z-test. This will help us determine if the difference in proportions is statistically significant.

1. We need to set up our null and alternative hypotheses:
   • Null hypothesis (H0): The proportion of normal weight men is equal to the proportion of normal weight women.
   • Alternative hypothesis (Ha): The proportion of normal weight men is not equal to the proportion of normal weight women.
2. We will calculate the test statistic using the formula:
   • Z = (p1 - p2) / sqrt(p * (1-p) * (1/n1 + 1/n2))
3. Next, we will compare the test statistic to the critical value at a significance level of 0.01. The critical value for a two-tailed test with a significance level of 0.01 is approximately ±2.576.
4. If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis and conclude that there is a significant difference in the proportions of normal weight men and women.

Answer 2

Based on the z-test, there is statistically significant evidence at the 0.01 level to conclude that there is a difference in the proportion of men and women who are of normal weight.

In this scenario, we're looking to determine if there's a statistically significant difference in the proportion of men and women who are of normal weight based on their BMI. To do this, we will conduct a hypothesis test for two proportions at the 0.01 significance level.

Here's what we're going to do step-wise:

1. State the Null and Alternative Hypotheses:

- Null Hypothesis
`(\( H_0 \))`: There is no difference in the proportion of men and women who are of normal weight. This can be written as
`\( p_m = p_w \)`, where
`\( p_m \)` is the proportion of men of normal weight and
`\( p_w \)` is the proportion of women of normal weight.

- Alternative Hypothesis
`(\( H_a \))`: There is a difference in the proportion of men and women who are of normal weight, which can be written as
`\( p_m \\eq p_w \)`.

2. Calculate the Test Statistic:

- We will use the formula for the z-test for two proportions, which is given by:

`\[ z = \frac{(\hat{p}_m - \hat{p}_w)}{\sqrt{\hat{p}(1-\hat{p})((1)/(n_m) + (1)/(n_w))}} \]`

- Where:

-
`\( \hat{p}_m = (203)/(750) \)` is the sample proportion of normal weight men.

-
`\( \hat{p}_w = (270)/(750) \)` is the sample proportion of normal weight women.

-
`\( \hat{p} \)` is the pooled proportion of normal weight individuals, calculated as
`\( \hat{p} = (203 + 270)/(750 + 750) \)`.

-
`\( n_m = 750 \)` is the number of men in the survey.

-
`\( n_w = 750 \)` is the number of women in the survey.

3. Determine the Critical Value:

- The critical value for a two-tailed test at the 0.01 level of significance can be found using a z-table or standard normal distribution table. Since it's a two-tailed test, we will be looking for the z-value that corresponds to an area of 0.005 in each tail (because 0.01 is split between the two tails).

4. Make a Decision:

- If the calculated z-value is greater than the critical z-value (in absolute terms), we reject the null hypothesis.

- If the calculated z-value is less than the critical z-value, we fail to reject the null hypothesis.

5. Interpret the Results:

- If we reject the null hypothesis, there is evidence to suggest there is a difference in the proportion of normal weight between men and women.

- If we fail to reject the null hypothesis, there isn't enough evidence to suggest a difference in the proportion of normal weight between men and women.

Let's go ahead and perform these calculations.

The calculated z-test statistic for the difference in proportions is approximately -3.723, and the critical z-value for a two-tailed test at the 0.01 level of significance is approximately 2.576. Since the absolute value of the z-statistic is greater than the critical value, we reject the null hypothesis.

Furthermore, the p-value of the test is approximately 0.000197, which is less than the significance level of 0.01. This also leads us to reject the null hypothesis.

Based on the z-test, there is statistically significant evidence at the 0.01 level to conclude that there is a difference in the proportion of men and women who are of normal weight.