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Answer :
(a) The p-value for x = 96.1 on cement mixture is approximately 0.1188. (b) The p-value for x= 100.8 is approximately 0.4283. (c) The p-value for x= 102.1 is approximately 0.3553.
To calculate the p-values for the observed statistics in the given hypothesis test, we will use the t-distribution since the population standard deviation is not known. Here's how we can calculate the p-values for each case:
(a) For x= 96.1: The observed statistic is x= 96.1. The sample size is n = 8.
First, we need to calculate the t-score:
t = (x - μ) / (s / √n)
where μ is the population mean, s is the sample standard deviation, and n is the sample size.
Here, μ = 101.2, s = 5.1, and n = 8:
t = (96.1 - 101.2) / (5.1 / √8) ≈ -1.775
Next, we need to find the p-value associated with this t-score in a t-distribution with (n - 1) degrees of freedom. In this case, we have (8 - 1) = 7 degrees of freedom.
Using a t-table or statistical software, we can find that the p-value for t ≈ -1.775 with 7 degrees of freedom is approximately 0.1188.
Therefore, the p-value for x= 96.1 is approximately 0.1188.
(b) For x= 100.8: The observed statistic is x= 100.8. The sample size is n = 8.
Using the same steps as above, we calculate the t-score:
t = (x - μ) / (s / √n)
Here, μ = 101.2, s = 5.1, and n = 8:
t = (100.8 - 101.2) / (5.1 / √8) ≈ -0.1961
Looking up the p-value for t ≈ -0.1961 with 7 degrees of freedom, we find that it is approximately 0.4283.
Therefore, the p-value for x= 100.8 is approximately 0.4283.
(c) For x= 102.1: The observed statistic is x= 102.1. The sample size is n = 8.
Again, following the same steps, we calculate the t-score:
t = (x - μ) / (s / √n)
Here, μ = 101.2, s = 5.1, and n = 8:
t = (102.1 - 101.2) / (5.1 / √8) ≈ 0.9902
Looking up the p-value for t ≈ 0.9902 with 7 degrees of freedom, we find that it is approximately 0.3553.
Therefore, the p-value for x= 102.1 is approximately 0.3553.
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