In 1960, census results indicated that the age at which men in a certain region first married had a mean of  23.2 years. It is widely suspected that young people today are waiting longer to get married. We want to find out if the mean age of first marriage has changed since then. Selecting a random sample 40 men who married for the first time last year at an average age of 24.1 years, with a standard deviation of 5.2 years what is teh t and p value

To determine if the mean age of first marriage has changed, we can perform a **one-sample t-test**. The hypothesis for this test would be:

- **Null Hypothesis (H₀):** The mean age of first marriage is still 23.2 years. \(\mu = 23.2\)
- **Alternative Hypothesis (H₁):** The mean age of first marriage has changed. \(\mu \neq 23.2\) (this is a two-tailed test).

We are given the following information:

- Sample mean (\(\bar{x}\)) = 24.1 years
- Population mean (\(\mu_0\)) = 23.2 years
- Sample standard deviation (\(s\)) = 5.2 years
- Sample size (\(n\)) = 40

### Step 1: Calculate the t-statistic

The formula for the t-statistic is:

\[
t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}
\]

Where:
- \(\bar{x}\) is the sample mean
- \(\mu_0\) is the population mean
- \(s\) is the sample standard deviation
- \(n\) is the sample size

Now, let me calculate that for you.The calculated t-statistic is approximately **1.095**.

### Step 2: Calculate the p-value

Since this is a two-tailed test, we need to find the p-value based on the t-statistic and the degrees of freedom (\(df = n - 1\)):

- Degrees of freedom: \( df = 40 - 1 = 39 \)

Let me now compute the p-value.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can calculate the p-value manually or using a calculator by looking up the cumulative distribution function (CDF) of the t-distribution with 39 degrees of freedom at the value of the t-statistic (1.095). This will give you the probability for one tail, and you would multiply it by 2 for the two-tailed test.

Would you like more details on how to do this or clarification on any of the steps?

Here are five related questions you might be interested in:
1. How do degrees of freedom affect the t-distribution?
2. What is the difference between a one-tailed and two-tailed test?
3. How can the t-distribution be used to assess the confidence interval for the population mean?
4. How would a larger sample size affect the t-statistic and p-value?
5. What are the assumptions underlying the one-sample t-test?

**Tip:** For small sample sizes, the t-distribution is used instead of the normal distribution because it accounts for more variability in the sample.

In 1960, census results indicated that the age at which men in a certain region first married had a mean of 23.2 years. It is widely suspected that young people today are waiting longer to get married. We want to find out if the mean age of first marriage has changed since then. Selecting a random sample of 40 men who married for the first time last year at an average age of 24.1 years, with a standard deviation of 5.2 years, what is the t and p value?

Learn how to perform a one-sample t-test to determine if the mean age of first marriage has changed since 1960. Using a sample of 40 men with a mean age of 24.1 years and a standard deviation of 5.2 years, calculate the t-statistic and p-value. Suitable for undergraduate statistics students.

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