Question

A simple random sample of size 28 has mean x=7.26. g The population standard deviation is σ=3.72. The population is normally distributed. Can you conclude that the population mean differs from 40? Determine the parameter to be tested.

Answer 1

Parameter tested: Population mean
`\mu`

We have enough evidence to conclude that the population mean for this case differs from 40.

Explanation:

1) Notation and definitions

n= 28 represent the sample size

`/bar x =7.26` represent the sample mean obtained

`\sigma =3.72` represent the population standard deviation known

`\mu_o =40` the value that we want to compare or test

2) Concepts and formulas to use

The system of hypothesis that we need to check for this case are

Null Hypothesis:
`\mu =40`

Alternative hypothesis:
`\mu \\eq 40`

We assume that the sample mean follows a normal distribution.

When conduct a Z test, to analyze if the population mean is equal to a apecified value
`\mu_o =40`:

In order to check the hypothesis we need to use the following statistic

`z=(\bar X -\mu_o)/((\sigma)/(√(n)))` (1)

A one sample test of means "compares the mean of a sample to a prespecified value and tests for a deviation from that value".

Check for the assumptions that he sample must satisfy in order to apply the test

• The dependent variable must be continuous (interval/ratio). Satisfied

• The observations are independent of one another. We assume it.

• The dependent variable should be approximately normally distributed. Satisfied

• The dependent variable should not contain any outliers. We assume it.

3) Calculate the statistic

Since we have all the info requires we can replace in formula (1) like this:

`z=(7.26 -40)/((3.72)/(√(28)))=-46.57`

4) Statistical decision

95% of the values in the normal standard distribution are between -1.96 and 1.96, if we obtain a value of z=-46.57, the p value for a two tailed test would be almost 0. And for this case at any significance level
`\alpha` we will reject the null hypothesis that the population mean is 40. So we have enough evidence to conclude that the population mean for this case differs from 40.