Question

You wish to test the following claim (HaHa) at a significance level of α=0.005α=0.005.

      Ho:μ=77.1Ho:μ=77.1
      Ha:μ<77.1Ha:μ<77.1

You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n=9n=9 with mean M=58.9M=58.9 and a standard deviation of SD=13.3SD=13.3.

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) αα
• greater than αα

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 less than 77.1.
• There is not sufficient evidence to warrant rejection of the claim that the population mean is less than 77.1.
• The sample data support the claim that the population mean is less than 77.1.
• There is not sufficient sample evidence to support the claim that the population mean is less than 77.1.

Answer 1

Since the population distribution is normal and we do not know the standard deviation, we will use the t-statistic here, which is given as

t= (xbar- mean)/ S/ sqrt(n)

where xbar= sample mean, mean= population mean, S= sample standard deviation, n= sample size

Now, t= (58.9- 77.1)/13.3/ sqrt(9)

= -18.2* sqrt(9)/13.3= -18.2* 3/ 13.3

= -54.6/13.3= -4.1052631579= -4.105 (upto three decimal places)

Again p-value for this t-score with left-tailed test ( since Ha: mean< 77.1) and n-1= 8 degrees of freedom, is given as 0.00170706= 0.0017( upto 4 decimal places)

Now, we see that the p-value= 0.0017 is less than alpha= 0.005.

Now, since the p-value is less than alpha, we will reject the null hypothesis and conclude that the test is Statistically significant.

The final conclusion will be that the sample data support the claim that the population mean is less than 77.1