Question

The average size of single-family homes in the U.S. is 2,390 square feet. A random sample of 100 homes in California yielded the mean of 2,507 square feet and the standard deviation of 257 square feet. Do the sample data provide sufficient evidence to conclude that the mean size of California homes exceeds the national average? Test using significance level of 0.01. (7 Points)

Answer 1

The decision rule is

Reject the null hypothesis

The conclusion is

There is sufficient evidence to conclude that the mean size of California homes exceeds the national average

Explanation:

From the question we are told that

The population mean is
`\mu = 2390 \ ft^2`

The sample size is n = 100

The sample mean is
`\= x = 2 507 \ ft^2`

The standard deviation is
`s = 257 \ ft^2`

The level of significance is
`\alpha = 0.01`

The null hypothesis is
`H_o : \mu = 2390`

The alternative hypothesis is
`H_a : \mu > 2390`

Generally the test statistics is mathematically represented as

`z = ( \= x - \mu )/( (s)/( √(n) ) )`

=>
`z = (2507 - 2390 )/( ( 257 )/( √(100 ) ) )`

=>
`z = 4.55`

From the z table the area under the normal curve to the left corresponding to 4.55 is

`p-value = P( X > 4.55 ) = 0.00`

From the value obtained we see that
`p-value < \alpha` hence

The decision rule is

Reject the null hypothesis

The conclusion is

There is sufficient evidence to conclude that the mean size of California homes exceeds the national average