Question

Suppose the average size of a new house built in a certain county in 2014 was 2,273 square feet. A random sample of 25 new homes built in this county was selected in 2018. The average square footage was 2,194​, with a sample standard deviation of 226 square feet. Complete parts a and b.

a. Using α =0.02​, does this sample provide enough evidence to conclude that the average house size of a new home in the county has changed since​ 2014?

Determine the null and alternative hypotheses.

H0​:μ ▼ greater than or equals ≥ not equals ≠ less than or equals ≤ equals =

H1​:μ ▼ less than < greater than > equals = not equals ≠ ​(Type integers or decimals. Do not​ round.)

Determine the appropriate critical value. Select the correct choice below and fill in the answer box within your choice. ​(Round to three decimal places as​ needed.)

A. tα/2 equals =

B. tα equals =

C. −tα equals =

Calculate the appropriate test statistic.

t-x= ​(Round to two decimal places as​ needed.)

State the conclusion. ▼

Reject/Do not reject H0.

There ▼ is/is not sufficient evidence to conclude that the average house size of a new home in the county has ▼ increased/stayed the same/changed/decreased since 2014.

b. Determine the precise​ p-value for this test using Excel.

The​ p-value is . ​(Round to three decimal places as​ needed.)

Answer 1

Solution:

Given:

the average size of a new house built in a certain county in 2014 was 2,273 square feet

Thus

Sample size = n = 25

Sample mean =

sample standard deviation = s = 226

Part a) Using α =0.02​, does this sample provide enough evidence to conclude that the average house size of a new home in the county has changed since​ 2014?

Part a-1) Determine the null and alternative hypotheses.

Since claim is non-directional , this is two tailed test.

H0​:μ = 2273

Vs

H1​:μ   ≠​ 2273

Part a-2) Determine the appropriate critical value.

Since this is two tailed test , find α / 2=0.02 / 2​ = 0.01

df = n - 1 = 25 -1 = 24

Thus look in t table for df = 24 and one tail area = α / 2 = 0.01 and find t critical value.

t_{α / 2} = 2.492

A. t_{α / 2} = 2.492

Part a-3) Calculate the appropriate test statistic.

Part a-4) State the conclusion.

Since absolute t test statistic value = 1.75 < t_{α / 2} = 2.492, we do not reject H0.

Thus correct answer is:
Do not reject H0.

There is not sufficient evidence to conclude that the average house size of a new home in the county has changed since 2014.

Part b)  Determine the precise​ p-value for this test using Excel.

=T.DIST.2T( absolute t test statistic , df)

=T.DIST.2T( 1.75 , 24)

=0.093

Thus p-value = 0.093