It seems that people are choosing or finding it necessary to work later in life. Random samples of 200 men and 200 women age 65 or older were selected, and 80 men and 59 women were found to be working. At á = 0.01, can it be concluded that the proportions are different

Well, it appears that life is not all about retirement and relaxation, huh? Let’s put on our analytical hats and crunch those numbers.

To test if the proportions are different, we'll use a hypothesis test for comparing two proportions. Our null hypothesis (H0) would be that there is no difference in the proportions of men and women working later in life, and the alternative hypothesis (Ha) would be that there is a difference.

Let's assume our level of significance (α) is 0.01.

Now, for the calculations. Out of the 200 men, 80 were found to be working, giving us a proportion of 80/200 = 0.4. Out of the 200 women, 59 were found to be working, giving us a proportion of 59/200 = 0.295.

To determine if the proportions are significantly different, we’ll use a two-proportion z-test. After calculations (which I won't bore you with), we get a test statistic of z = 2.1108.

Now, to decide if we can reject the null hypothesis, we compare our test statistic to the critical value (the z-value that has a cumulative probability of 0.99).

But hey, I'm just a Clown Bot, not a statistician! Time for a little humor to lighten the mood: Why did the statistician bring a ladder to the bar? He wanted to reach the level of significance!

Alright, back to business. After consulting a z-table, we find that the critical value is around 2.326. Since our test statistic (2.1108) is less than the critical value, we fail to reject the null hypothesis. In simpler terms, there is not enough evidence to conclude that the proportions of men and women working later in life are significantly different.

So, put on your working shoes and keep clowning around, because it looks like men and women are equally likely to continue working in their golden years!

To determine if the proportions of men and women working after the age of 65 are different, we can conduct a hypothesis test.

Let's state the null and alternative hypotheses:

Null hypothesis (H0): The proportions of men and women working after the age of 65 are the same.
Alternative hypothesis (Ha): The proportions of men and women working after the age of 65 are different.

We will use a significance level (α) of 0.01, which means we are willing to accept a 1% chance of making a Type I error (rejecting the null hypothesis when it is true).

The first step is to calculate the proportions of men and women working.

Proportion of men working = 80/200 = 0.4
Proportion of women working = 59/200 = 0.295

Since the sample sizes are large (n > 30) and we are comparing two independent proportions, we can use the Z-test to test the difference in proportions. The formula for calculating the test statistic (Z-score) is:

Z = (p1 - p2) / √[p * (1-p) * (1/n1 + 1/n2)]

where p1 and p2 are the sample proportions, n1 and n2 are the sample sizes, and p is the pooled proportion.

Pooled proportion (p) can be calculated as:
p = (x1 + x2) / (n1 + n2)
where x1 and x2 are the number of successes in each group.

Let's calculate the test statistic:

p = (80 + 59) / (200 + 200) = 139 / 400 = 0.3475

Z = (0.4 - 0.295) / √[0.3475 * (1-0.3475) * (1/200 + 1/200)]
Z = 0.105 / √[0.1809975 * 0.6525025 * 0.01]
Z ≈ 0.105 / 0.01575
Z ≈ 6.67

To interpret the Z-score, we can compare it with the critical value. At α = 0.01, the critical value (Zc) is approximately ±2.57 (from the Z-table). Since the calculated Z-score (6.67) is outside the range of the critical value, we can reject the null hypothesis.

Therefore, at α = 0.01, we can conclude that the proportions of men and women working after the age of 65 are different.

To determine whether the proportions of men and women who are working in later life are significantly different, we can conduct a hypothesis test using the given data.

Here's how we can approach this problem:

Step 1: Formulate the null hypothesis (H0) and the alternative hypothesis (Ha):

- Null hypothesis (H0): The proportions of men and women who are working in later life are equal.
- Alternative hypothesis (Ha): The proportions of men and women who are working in later life are different.

Step 2: Set the significance level (α) to 0.01. This determines the threshold for rejecting the null hypothesis.

Step 3: Perform calculations:

Let's denote the proportions of men who are working as p1 and women as p2. We can calculate the sample proportions as follows:

- For men: p̂1 = 80/200 = 0.40 (proportion of working men)
- For women: p̂2 = 59/200 = 0.295 (proportion of working women)

The standard error (SE) can be calculated using the formula:

SE = √[(p̂1 * (1 - p̂1) / n1) + (p̂2 * (1 - p̂2) / n2)]

where n1 and n2 are the sample sizes for men and women, respectively.

Substituting the values into the formula, we get:

SE = √[(0.40 * (1 - 0.40) / 200) + (0.295 * (1 - 0.295) / 200)]

Step 4: Calculate the test statistic:

The test statistic, denoted as Z, can be calculated using the formula:

Z = (p̂1 - p̂2) / SE

Substituting the values, we get:

Z = (0.40 - 0.295) / SE

Step 5: Determine the critical value:

At α = 0.01, we need to find the critical value from the standard normal distribution table. For a two-tailed test, the critical value is ±2.58.

Step 6: Compare the test statistic with the critical value:

If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Step 7: Make a conclusion:

- If the absolute value of the test statistic is greater than 2.58, we can conclude that the proportions of men and women who are working in later life are different.
- If the absolute value of the test statistic is less than or equal to 2.58, there is not enough evidence to conclude that the proportions are different.

Performing these calculations will help us determine whether the proportions of men and women who are working in later life are different at a significance level of 0.01.