In the 2000 census, the so-called “long form”
received by one of every six households contained
52 questions, ranging from your occupation and
income all the way to whether you had a bathtub.
According to the U.S. Census Bureau, the mean
completion time for the long form is 38 minutes.
Assuming a standard deviation of 5 minutes and a
simple random sample of 50 persons who filled out
the long form, what is the probability that their
average time for completion of the form was more than
45 minutes?

To solve this problem, we will apply the **Central Limit Theorem (CLT)**, which allows us to approximate the sampling distribution of the sample mean. Since the sample size \( n = 50 \) is sufficiently large, the sampling distribution of the sample mean will be approximately normal, even if the population distribution is not.

### Given information:
- Mean completion time \( \mu = 38 \) minutes.
- Standard deviation \( \sigma = 5 \) minutes.
- Sample size \( n = 50 \).
- We are asked to find the probability that the sample mean is more than 45 minutes.

We can calculate the probability that the sample mean is more than 45 minutes using the standard normal distribution.

### Step 1: Calculate the standard error (SE) of the sample mean
The standard error of the sample mean is given by:

\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{5}{\sqrt{50}} \approx 0.7071
\]

### Step 2: Convert the sample mean of 45 minutes to a z-score
The z-score tells us how many standard deviations 45 minutes is from the population mean. The formula for the z-score is:

\[
z = \frac{\bar{x} - \mu}{SE} = \frac{45 - 38}{0.7071} = \frac{7}{0.7071} \approx 9.9
\]

### Step 3: Find the probability
Using the z-score of 9.9, we can find the corresponding probability from the standard normal distribution table (or using a calculator). A z-score of 9.9 is extremely high, meaning the probability is virtually 0.

\[
P(\bar{x} > 45) \approx 0
\]

### Conclusion:
The probability that the average completion time of the form was more than 45 minutes is practically **0**.

Do you need any further details or have any questions? Here are 5 related questions for deeper understanding:

1. What is the Central Limit Theorem, and why is it important in this problem?
2. How does increasing the sample size affect the standard error of the sample mean?
3. What happens to the z-score if the standard deviation was larger?
4. Can you explain the significance of a z-score of 9.9?
5. What if the mean time for a different group was 42 minutes? How would the probability change?

**Tip:** When calculating probabilities involving sample means, always check if the sample size is large enough to apply the Central Limit Theorem.

In the 2000 census, the so-called “long form” received by one of every six households contained 52 questions, ranging from your occupation and income all the way to whether you had a bathtub. According to the U.S. Census Bureau, the mean completion time for the long form is 38 minutes. Assuming a standard deviation of 5 minutes and a simple random sample of 50 persons who filled out the long form, what is the probability that their average time for completion of the form was more than 45 minutes?

Learn how to calculate the probability that the average completion time for the 2000 census long form was more than 45 minutes. This detailed solution involves key statistical concepts such as the Central Limit Theorem, sampling distribution, and z-scores. Suitable for college-level students.

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