The heat evolved in calories per gram of a cement mixture is approximately normally distributed. The mean is thought to be 100.4, and the standard deviation is 4.2. We wish to test [tex]H_0: \mu = 100.4[/tex] versus [tex]H_1: \mu \neq 100.4[/tex] with a sample of [tex]n = 9[/tex] specimens. Calculate the P-value if the observed statistic is:

(a) [tex]\bar{x} = 96.5[/tex]
(b) [tex]\bar{x} = 100.8[/tex]
(c) [tex]\bar{x} = 103.6[/tex]

Round your answers to four decimal places.

To calculate the p-value, we use the t-distribution and the formula t = (X- μ) / (s / √n). We calculate the t-value for three different observed statistics and find the corresponding p-values.

Explanation:

To calculate the p-value, we need to use the t-distribution because the population standard deviation is unknown and we have a sample size of n=9. First, we calculate the t-value using the formula:

t = (X - μ) / (s / √n)

Using the given data, we can calculate the t-values as follows:

(a) t = (96.5 - 100.4) / (4.2 / √9) = -3.57

(b) t = (100.8 - 100.4) / (4.2 / √9) = 0.95

(c) t = (103.6 - 100.4) / (4.2 / √9) = 1.33

Next, we find the p-value associated with each t-value using the t-distribution table or a calculator. The p-values are:

(a) p-value ≈ 0.0061

(b) p-value ≈ 0.1800

(c) p-value ≈ 0.1070