As part of your analysis for the Real Estate Advisory Board, you are investigating whether there is a statistically significant difference in average market prices between houses and apartments.You collect a random sample of recent sales and obtain the following summary statistics:- Mean price of houses: $981021- Standard deviation of house prices: $50859- Sample size for houses: 35- Mean price of apartments: $702945- Standard deviation of apartment prices: $34275- Sample size for apartments: 40You want to test the following hypotheses:- H₀: μ₁ = μ₂ (no difference in means)- H₁: μ₁ ≠ μ₂ (difference in means)Assume unequal population variances, and use the Welch’s t-test. Hint: Avoid any rounding of intermediate steps, and use full decimals. Only round the final result to 3 decimal places Enter the sample value of the test statistic (t):Answer Question 10[input]

Question

BlackTom AI · Accepted Answer

We begin by restating the given data and the goal of the test.
- Mean price of houses (group 1): 981,021; standard deviation s1 = 50,859; sample size n1 = 35.
- Mean price of apartments (group 2): 702,945; standard deviation s2 = 34,275; sample size n2 = 40.
- Hypotheses: H0: μ1 = μ2 vs H1: μ1 ≠ μ2, using Welch’s t-test ( unequal variances ).

To compute the Welch t-statistic, we use:
t = (mean1 − mean2) / sqrt( s1^2/n1 + s2^2/n2 )

Step 1: Compute the variance terms divided by sample sizes.
- s1^2/n1 = (50,859^2) / 35. First compute s1^2: 50,859 × 50,859 ≈ 2,586,637,881. Divide by 35: 2,586,637,881 / 35 ≈ 73,903,939.46.
- s2^2/n2 = (34,275^2) / 40. Compute s2^2: 34,275 × 34,275 = 1,174,775,625. Divide by 40: 1,174,775,625 / 40 = 29,369,390.625.

Step 2: Sum the two variance terms for the Welch denominator.
- s1^2/n1 + s2^2/n2 ≈ 73,903,939.46 + 29,369,390.625 ≈ 103,273,330.085.

Step 3: Take the square root to obtain the standard error of the difference in means under Welch’s approach.
- SE = sqrt(103,273,330.085) ≈ 10,164 (more precisely around 10,161–10,164 depending on rounding; using the precise sum gives ~10,161.9 to ~10,164.0).

Step 4: Compute the difference in sample means.
- mean1 − mean2 = 981,021 − 702,945 = 278,076.

Step 5: Compute the t-statistic.
- t ≈ 278,076 / 10,164 ≈ 27.36.

Remark on precision: The instruction asks to avoid rounding intermediate steps and to round the final result to 3 decimal places. Using the more precise SE from the exact sum of s1^2/n1 and s2^2/n2 yields t ≈ 27.363 (to three decimals). The key is that the computed t is extremely large in magnitude, indicating a very strong difference between means under the Welch formulation.

Summary: The sample value of the test statistic t is approximately 27.363 when computed with full precision for the intermediate steps and final rounding to three decimals.

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