Question

According to the census bureau, 3.29 people reside in the typical american household. a sample of 16 households in arizona retirement communities showed the mean number of residents per household was 2.76 residents. the standard deviation of this sample was 1.29 residents. at the .05 significance level, is it reasonable to conclude the mean number of residents in the retirement community household is less than 3.29 persons?

Answer 1

We conclude that the mean number of residents in the retirement community household is more than or equal to 3.29 persons.

Explanation:

We are given that according to the census bureau, 3.29 people reside in the typical american household.

A sample of 16 households in Arizona retirement communities showed the mean number of residents per household was 2.76 residents. the standard deviation of this sample was 1.29 residents.

Let
`\mu` = mean number of residents in the retirement community household

So, Null Hypothesis,
`H_0` :
`\mu \geq` 3.29 persons {means that the mean number of residents in the retirement community household is more than or equal to 3.29 persons}

Alternate Hypothesis,
`H_A` :
`\mu` < 3.29 persons {means that the mean number of residents in the retirement community household is less than 3.29 persons}

The test statistics that will be used here is One-sample t test statistics as we don't know about population standard deviation;

T.S. =
`(\bar X-\mu)/((s)/(√(n) ) )` ~
`t_n_-_1`

where,
`\bar X` = sample mean number of residents per household = 2.76

s = sample standard deviation = 1.29 residents

n = sample of households = 16

So, test statistics =
`(2.76-3.29)/((1.29)/(√(16) ) )` ~
`t_1_5`

= -1.643

The value of the test statistics is -1.643.

Now at 0.05 significance level, the t table gives critical value of -1.753 at 15 degree of freedom for left-tailed test. Since our test statistics is more than the critical values of t as -1.643 > 1.753, so we have insufficient evidence to reject our null hypothesis as it will not fall in the rejection region due to which we fail to reject our null hypothesis.

Therefore, we conclude that the mean number of residents in the retirement community household is more than or equal to 3.29 persons.