To solve this problem, let's break down the details first:

We are conducting a two-tailed hypothesis test to check whether the mean home price is different from $250K or not. The test statistic $t$ was calculated as 3.139342 with 11 degrees of freedom (since the sample size is 12, degrees of freedom = 12 - 1 = 11).

Important Considerations:

• This is a two-tailed test, so we will need to calculate the p-value for both tails of the t-distribution.
• The Excel commands should correctly calculate the two-tailed p-value, which is $2 \times \text{P}(T > t)$ for a right-tailed value, or equivalently $2 \times \text{P}(T < -t)$ for a left-tailed value.

Let's review the given options:

1. =2*T.DIST.RT(3.139342,11)

   • T.DIST.RT(t, df) calculates the right-tail probability for the t-distribution.
   • Multiplying by 2 gives the two-tailed p-value, so this is correct for a two-tailed test.
     (Correct)

2. =2*(1-T.DIST(3.139342,11, TRUE))

   • T.DIST(t, df, TRUE) calculates the cumulative distribution function (CDF), or the area to the left of the given t-value.
   • $1 - \text{CDF}(t)$ gives the right-tail probability.
   • Multiplying by 2 makes it a two-tailed test. Hence, this formula is also correct.
     (Correct)

3. =2*T.DIST(-3.139342,11, TRUE)

   • T.DIST(-t, df, TRUE) calculates the area to the left of a negative t-value.
   • In a symmetric t-distribution, the area to the left of $-t$ is the same as the area to the right of $+t$.
   • So multiplying this by 2 gives the two-tailed p-value.
     (Correct)

4. =2*T.DIST(3.139342,11, TRUE)

   • T.DIST(t, df, TRUE) gives the area to the left of the t-value.
   • Multiplying this by 2 would give an incorrect p-value because it includes both the left-tail and the right-tail probability for positive t.
   • This is incorrect for a two-tailed test.
     (Incorrect)

Correct Selections:

• =2*T.DIST.RT(3.139342,11)
• =2*(1-T.DIST(3.139342,11, TRUE))
• =2*T.DIST(-3.139342,11, TRUE)

Would you like more details or have any questions about this explanation?

Here are 5 related questions:

1. How do you interpret a p-value in hypothesis testing?
2. What is the relationship between degrees of freedom and the shape of the t-distribution?
3. What is the difference between one-tailed and two-tailed tests in hypothesis testing?
4. How do you calculate the test statistic for a t-test?
5. When should you use a t-test versus a z-test in hypothesis testing?

Tip: When conducting a two-tailed test, always remember to multiply the one-tailed probability by 2 to get the correct p-value.