Calculating P-value and Acceptance region when given some small sample of data

Question

The right-hand strength in pounds was measured and data is as follows

29.4  30.8  30.6  31.5  32.1  31.7  30.3  30.8

a. What is P-value for rejecting the null hypotheses that the population mean is equal to 32?

b. Find acceptance region in testing hypotheses in (a). with a significance level (alpha) equal to 0.1. (Assume normality in both)

Answer 1

Find a discussion of one sample t tests in your text or class notes. And I suppose you are using some kind of software in your course.

How about looking to see what happens when these (claimed normal) observations are used in a t test in R (or some other statistical program or using a statistical calculator).

x = c(29.4, 30.8, 30.6, 31.5, 32.1, 31.7, 30.3, 30.8)
t.test(x, mu = 32)

        One Sample t-test

data:  x
t = -3.6238, df = 7, p-value = 0.008468
alternative hypothesis: true mean is not equal to 32
95 percent confidence interval:
 30.18222 31.61778
sample estimates:
mean of x 
     30.9 

The P-value is given in the output. When the t statistic has DF = 7, the probability of a smaller value than 30.9 is $P(T < -3.6238) = 0.00423.$ But this is a 2-sided test (Why do I say that?), so you have to consider the probability of a value at least as far into the upper tail. So the P-value is $0.00847$ as in the output.

pt(-3.6238, 7)
[1] 0.004233656
2 * pt(-3.6238, 7)
[1] 0.008467311

The critical value $c$ for a test at the $10\%$ level is the probability that $|T| \ge c,$ again with DF = 7. You should be able to use use a printed table of percentage point or quantiles of t distributions to find the value that cut 5% from each tail of that t distribution. Look on row DF = 7 of such a table and see if you can find something close to $c = 1.895.$

qt(.95, 7) 
[1] 1.894579

Notice that you need the actual data (or at least $\bar X$ and $S_X$ in order to find the P-value, but to find $c.$ you need only know that you have $n = 8$ normally distributed observations.

Now match this whole answer with examples in your text or notes.