Question

You wish to test the following claim (Ha) at a significance level of α=0.02.

      Ho:μ=51.8
      Ha:μ≠51.8

You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n=22 with mean M=49 and a standard deviation of SD=5.7.

What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic =

What is the p-value for this sample? (Report answer accurate to four decimal places.)
p-value =

The p-value is...

less than (or equal to) α

greater than α

This test statistic leads to a decision to...

reject the null

accept the null

fail to reject the null

As such, the final conclusion is that...

There is sufficient evidence to warrant rejection of the claim that the population mean is not equal to 51.8.

There is not sufficient evidence to warrant rejection of the claim that the population mean is not equal to 51.8.

The sample data support the claim that the population mean is not equal to 51.8.

There is not sufficient sample evidence to support the claim that the population mean is not equal to 51.8.

Answer 1

Solution:

   We are given that: the population is normally distributed with unknown standard deviation.

A sample of size n=22 with mean and a standard deviation of SD= s = 5.7

We wish to test:

Vs

Part i) What is the test statistic for this sample?

Since population standard deviation is unknown and sample size n = 22 is small, we use t test statistic.

Part ii) What is the p-value for this sample?

We need to use either Excel or TI 84 plus calculator to find p - value.

In Excel we use following command:

=T.DIST.2T( x , df )

where x = t test statistic value = -2.304

and df = n   - 1= 22 - 1 = 21

Thus

=T.DIST.2T( 2.304 , 21 )                    ( use absolute t value in excel)

=0.0315

Thus p-value = 0.0315

In TI 84plus calculator we use following steps:

1) Press 2ND

2) Press VARS

3) Select tcdf(

then enter numbers:

for negative sign , press (-) button which is in the bottom of calculator.

Then click on Paste and press Enter two times

Then multiply the answer by 2 to get p-value:

Thus p-value = 0.0315

Part iii) p value is ....?

p value = 0.0315 > significance level.

Thus The p-value is greater than α.

Part iv) This test statistic leads to a decision to fail to reject the null

Since p value > α.

Part v) As such, the final conclusion is that:

Here claim is Ha. Since we failed to reject null hypothesis, we conclude that: There is not sufficient sample evidence to support the claim that the population mean is not equal to 51.8.