Assume that home prices are normally distributed. A realtor claims that the average home price in a certain neighborhood is $250K. You think the price is higher. A sample of 12 recent home sales gives an average price of $279K with a standard deviation of $32K. What is the t-test statistic for your hypothesis? Round your answer to three decimal places. Recall: 𝑡=𝑋¯−𝜇0S.E.(𝑋¯)



We are tasked with calculating the t-test statistic for a one-sample t-test. The formula for the t-test statistic is:

\[
t = \frac{\bar{X} - \mu_0}{S.E.(\bar{X})}
\]

Where:
- $\bar{X}$ is the sample mean,
- $\mu_0$ is the population mean under the null hypothesis,
- $S.E.(\bar{X})$ is the standard error of the mean.

### Step 1: Define the known values:
- Sample mean ($\bar{X}$) = 279,000
- Population mean under the null hypothesis ($\mu_0$) = 250,000
- Sample standard deviation (s) = 32,000
- Sample size (n) = 12

### Step 2: Calculate the standard error of the mean ($S.E.(\bar{X})$):
The formula for the standard error of the mean is:

\[
S.E.(\bar{X}) = \frac{s}{\sqrt{n}}
\]

Substitute the known values:

\[
S.E.(\bar{X}) = \frac{32,000}{\sqrt{12}} \approx \frac{32,000}{3.464} \approx 9,236.353
\]

### Step 3: Calculate the t-test statistic:
Now, substitute the values into the formula for the t-test statistic:

\[
t = \frac{279,000 - 250,000}{9,236.353} = \frac{29,000}{9,236.353} \approx 3.140
\]

Thus, the t-test statistic is approximately **3.140**.

Would you like further details on how to interpret this result, or any additional explanation?

Here are five related questions:
1. How do you interpret the t-test statistic in terms of the null hypothesis?
2. What is the critical value of t for a significance level of 0.05 with 11 degrees of freedom?
3. How does increasing the sample size affect the standard error and the t-test statistic?
4. What assumptions are made when performing a t-test on this data?
5. How would the result change if the standard deviation were smaller?

**Tip**: A higher t-test statistic indicates stronger evidence against the null hypothesis.

Assume that home prices are normally distributed. A realtor claims that the average home price in a certain neighborhood is $250K. You think the price is higher. A sample of 12 recent home sales gives an average price of $279K with a standard deviation of $32K. What is the t-test statistic for your hypothesis? Round your answer to three decimal places.

Calculate the t-test statistic for the hypothesis that average home prices in a neighborhood exceed $250,000. With a sample mean of $279,000 and a standard deviation of $32,000 based on 12 sales, this example illustrates the t-test for comparing sample means. Ideal for undergraduate students studying statistics.

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