Question

Consider the following hypothesis test:

H0: µ = 20
Ha: µ < 20
A sample of 50 provided a sample mean of 19.4. The population standard deviation is 2.
a. Compute the value of the test statistic (to 2 decimals).
b. What is the p-value (to 3 decimals)?
c. Using a = .05, can it be concluded that the population mean is less than 20?
d. Using a = .05, what is the critical value for the test statistic?
e. State the rejection rule: Reject H0 if z is the critical value.
f. Using a = .05, can it be concluded that the population mean is less than 20?

Answer 1

a

`z = - 2.12`

b

`p- value = 0.017003`

c

There is sufficient evidence to show that the population mean is less than 20

d

`z_(0.05) =- 1.645`

e

The decision rule is

Reject the null hypothesis

f

The conclusion is

There is sufficient evidence to show that the population mean is less than 20.

Explanation:

From the question we are told that

The null hypothesis is
`H_o: \mu = 20`

The alternative hypothesis is Ha: µ < 20

The sample size is n = 50

The sample mean is
`\= x = 19.4`

The level of significance is
`\alpha = 0.05`

The population standard deviation is
`\sigma = 2`

Generally the test statistics is mathematically represented as

`z = (\= x - \mu )/( (\sigma )/( √(n) ) )`

=>
`z = (19.4 - 20 )/( (2)/( √(50 ) ) )`

=>
`z = - 2.12`

From the z table the area under the normal curve to the left corresponding to -2.12 is

`(P< -2.12) = 0.017003`

Generally the p-value is

`p- value = 0.017003`

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

The decision rule is

Reject the null hypothesis

The conclusion is

There is sufficient evidence to show that the population mean is less than 20.

Generally form the normal distribution table the critical value at a level of significance of
`\alpha = 0.05` is

`z_(0.05) =- 1.645`

Generally given that
`z < z_(0.05)`

The decision rule is

Reject the null hypothesis

The conclusion is

There is sufficient evidence to show that the population mean is less than 20.