The General Social Survey asked 1308 people whether they performed any volunteer work during the past year. A total of 519 people said they did.

Question

(a) Find a point estimate for the proportion of people who performed volunteer work during the past year. Round the answer to at least three decimal places.

us a TI-84 calculator

Answer 1

To find the point estimate for the proportion of people who performed volunteer work during the past year, you can use the formula for the sample proportion:

\[ \hat{p} = \frac{x}{n} \]

where:

• \(\hat{p}\) is the sample proportion,
• \(x\) is the number of people who performed volunteer work, and
• \(n\) is the total number of people surveyed.

In this case:

• \(x = 519\) (the number of people who performed volunteer work),
• \(n = 1308\) (the total number of surveyed people).

So the calculation would be:

\[ \hat{p} = \frac{519}{1308} \]

Now calculating this:

\[ \hat{p} = 0.397 \]

Thus, the point estimate for the proportion of people who performed volunteer work during the past year is approximately 0.397 when rounded to three decimal places.

Answer 2

(b) Construct a 99.5% confidence interval for the proportion of people who performed volunteer work during the past year. Round the answer to at least three decimal places.

Answer 3

To construct a confidence interval for the proportion of people who performed volunteer work, we use the formula for the confidence interval for a population proportion:

\[ \hat{p} \pm z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]

Step 1: Calculate \(\hat{p}\)

From the previous calculation, we have:

\[ \hat{p} = 0.397 \]

Step 2: Find the critical value \(z\) for a 99.5% confidence level

For a 99.5% confidence interval, the significance level \(\alpha\) is:

\[ \alpha = 1 - 0.995 = 0.005 \]

Since we are looking for the z-value corresponding to the upper 0.0025 tail (because we need half the alpha for a two-tailed test), we can find \(z\) using a standard normal distribution table or calculator. The critical value for \(z\) at 99.5% is approximately:

\[ z \approx 2.807 \]

Step 3: Calculate the standard error (SE)

Now, we can calculate the standard error (SE):

\[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.397(1 - 0.397)}{1308}} \]

Calculating \(1 - \hat{p}\):

\[ 1 - \hat{p} = 1 - 0.397 = 0.603 \]

Now plug it into the SE formula:

\[ SE = \sqrt{\frac{0.397 \times 0.603}{1308}} \approx \sqrt{\frac{0.239991}{1308}} \approx \sqrt{0.0001834} \approx 0.01355 \]

Step 4: Calculate the margin of error (ME)

The margin of error (ME) is given by:

\[ ME = z \cdot SE \approx 2.807 \cdot 0.01355 \approx 0.0380 \]

Step 5: Construct the confidence interval

Now we can construct the confidence interval:

\[ (\hat{p} - ME, \hat{p} + ME) = (0.397 - 0.0380, 0.397 + 0.0380) = (0.359, 0.435) \]

Final Answer

The 99.5% confidence interval for the proportion of people who performed volunteer work during the past year is:

\[ (0.359, 0.435) \]

When rounded to three decimal places, the interval is approximately:

\[ (0.359, 0.435) \]

Answer 4

Can you explain step 2 better

Answer 5

Certainly! Step 2 involves finding the critical value \( z \) for the desired confidence level of 99.5%. This critical value is important because it helps determine how wide the confidence interval will be. Here’s a detailed explanation:

Understanding the Confidence Level

1. Confidence Level and Alpha:

   • A 99.5% confidence level means we expect 99.5% of the intervals created from repeated samples to contain the true population proportion.
   • The significance level \(\alpha\) is the probability that the true parameter is not captured in the confidence interval. For a 99.5% confidence level:
     • \(\alpha = 1 - 0.995 = 0.005\)

2. Dividing Alpha:

   • Since we are constructing a two-tailed confidence interval (considering both directions away from the sample proportion), we split \(\alpha\) into two tails:
     • \(\alpha/2 = 0.005/2 = 0.0025\)

3. Finding the Critical Value \( z \):

   • We need to find the \( z \)-value that corresponds to the upper tail area of \( 0.0025\).
   • In a standard normal distribution, this \( z \)-value tells us how many standard deviations away from the mean we need to go in order to capture the central 99.5% of the distribution.

Using the Z-Table or Calculator

To find this \( z \)-value, you can either:

1. Use a Z-table:

   • Look up \( 1 - 0.0025 = 0.9975 \) in the Z-table (which provides the area to the left of the \( z \)-value). The closest value to \( 0.9975 \) will give you the corresponding \( z \)-value.
   • This value is typically found to be approximately \( z = 2.807 \).

2. Use a Calculator:

   • Many graphing calculators or statistical software have functions to directly compute the critical \( z \)-value for a given confidence level or area. For example, on a TI-84 calculator, you can use the invNorm function:
     • Input the value that corresponds to the left side of the z-distribution:
       • invNorm(0.9975) will yield the critical z-value \( z \approx 2.807 \).

Conclusion

The critical value \( z \) you found in this step represents the number of standard deviations you must extend from the sample proportion to create the confidence interval. This value is then used in the margin of error calculation, which completes the construction of the confidence interval.

With a critical value \( z \approx 2.807\), you have enough information to calculate the margin of error and thereby construct your confidence interval around the point estimate \(\hat{p}\).