Question

A real estate company is interested in testing whether the mean time that families in Gotham have been living in their current homes is less than families in Metropolis. Assume that the two population variances are equal. A random sample of 100 families from Gotham and a random sample of 150 families in Metropolis yield the accompanying data on length of residence in current homes.
What is the estimated standard error of the difference between the 2 sample​ means? Round to the nearest hundredth.
​Gotham: XG = 35 ​months, SG2 = 900 ​
Metropolis: XM = 50 ​months, SM2 = 1050

Answer 1

Answer:

Step-by-step explanation:

Hello!

The objective is to compare if the average time that families live in Gotham and Metropolis, the real state company thinks the average time is less in Gotham than in Metropolis. There are two populations of interest and therefore two variables:

X₁: Time a family has lived in Gotham.

n₁= 100 families

X[bar]₁= 35 months

S₁= 900 days

X₂: Time a family has lived in Metropolis.

n₂= 150 families

X[bar]₂= 50 months

S₂= 1050 days

You are studying the population means of both variables, so you have to work using the distribution of the sample means:

If:

X[bar]₁~N(μ₁;σ₁²/n₁)

X[bar]₂~N(μ₂;σ₂²/n₂)

We can say that the difference between these both distributions will have the following distribution:

(X[bar]₁-X[bar]₂)~N(μ₁-μ₂;σ₁²/n₁+σ₂²/n₂)

The population variance for the difference between the two sample means is:

σ₁²/n₁+σ₂²/n₂

Since both σ₁² = σ₂² = σ² we can say that:

σ² (1/n₁+1/n₂)To study the difference between two normal populations with unknown bu equal population variances the distribution to use is:

$Z= \frac{(Xbar_1-Xbar_2)-(Mu_1-Mu_2)}{Sa*\sqrt{\frac{1}{n_1} +\frac{1}{n_2} } } ~~N(0;1)$

The estimation of the standard error of the difference is:

$Sa*\sqrt{\frac{1}{n_1} +\frac{1}{n_2} }$

$Sa^2= \frac{(n_1-1)S_1^2+(n_2-1)S_2^2}{n_1+n_2-2} = \frac{(99*900^2)+(149*1050^2)}{100+150-2}= 985735.8871$

Sa= 992.8423≅ 992.84

$992.84*\sqrt{\frac{1}{100} +\frac{1}{150} }= 128.1751= 128.18$

The standard error of the difference between the two sample means is 128.18.

I hope this helps!