Question

According to the Census Bureau, 3.36 people reside in the typical American household. A sample of 25 households in Arizona retirement communities showed the mean number of residents per household was 2.71 residents. The standard deviation of this sample was 1.10 residents. At the .10 significance level, is it reasonable to conclude the mean number of residents in the retirement community household is less than 3.36 persons? State the null hypothesis and the alternate hypothesis

Answer 1

We conclude that the mean number of residents in the retirement community household is less than 3.36 persons.

Explanation:

We are given the following in the question:

Population mean, μ = 3.36

Sample mean,
`\bar{x}` = 2.71

Sample size, n = 25

Alpha, α = 0.10

Sample standard deviation, s = 1.10

First, we design the null and the alternate hypothesis

`H_(0): \mu = 3.36\text{ residents per household}\\H_A: \mu < 3.36\text{ residents per household}`

We use One-tailed(left) t test to perform this hypothesis.

Formula:

`t_(stat) = \displaystyle\frac{\bar{x} - \mu}{(s)/(√(n)) }` Putting all the values, we have

`t_(stat) = \displaystyle(2.71 - 3.36)/((1.10)/(√(25)) ) = -2.95`

Now,
`t_(critical) \text{ at 0.10 level of significance, 24 degree of freedom } =-1.31`

Since,

`t_(stat) < t_(critical)`

We reject the null hypothesis and fail to accept it.

Thus, we conclude that the mean number of residents in the retirement community household is less than 3.36 persons.