A machine in the student lounge dispenses coffee. The average cup of coffee is supposed to contain 7.0 ounces. A random sample of nine cups of coffee from this machine show the average content to be 7.2 ounces with a standard deviation of 0.40 ounce. Do you think that the machine has slipped out of adjustment and that the average amount of coffee per cup is different from 7 ounces? Use a 5% level of significance.
State the null and alternate hypotheses.
Null Hypothesis (H0): µ=7.0
Alternate Hypothesis (Ha): µ≠7.0
What sampling distribution will you use? What assumptions are you making?
The standard normal, since we assume that x has a normal distribution with known 𝜎.
The Student's t, since we assume that x has a normal distribution with unknown 𝜎.
The Student's t, since n is large with unknown 𝜎.
The standard normal, since we assume that x has a normal distribution with unknown 𝜎.
What is the value of the sample test statistic? (Round your answer to three decimal places.)
t = (x̄-µ)/(s/√n) = (7.2-7.0)/(0.40/√9) = 1.5
Estimate the P-value.
Using a t-distribution table, we look down the row of α=0.05 and df=n-1=9-1=8. Our t-statistic of t=1.5 from before is between 1.397 and 1.860 for a one-tailed test, which is in-between p-values of 0.05 and 0.1, but let's call is 0.075. Because this is a two-tailed test, we just simply double our in-between p-value to get 0.15, which means the 3rd choice is correct.
P-value > 0.500
0.250 < P-value < 0.500
0.100 < P-value < 0.250
0.050 < P-value < 0.100
0.010 < P-value < 0.050
P-value < 0.010
Sketch the sampling distribution and show the area corresponding to the P-value.
Hopefully this should be intuitive enough. If not, look up example distribution graphs.
Will you reject or fail to reject the null hypothesis? Are the data statistically significant at level 𝛼?
If our estimated p-value is 0.15, which is greater that α=0.05, then we would fail to reject the null hypothesis and conclude the data are not statistically significant, which means the 2nd option is correct
At the 𝛼 = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.
At the 𝛼 = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.
At the 𝛼 = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the 𝛼 = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
Interpret your conclusion in the context of the application.
There is sufficient evidence at the 0.05 level to conclude that the mean amount of coffee per cup differs from 7 ounces.
There is insufficient evidence at the 0.05 level to conclude that the mean amount of coffee per cup differs from 7 ounces.
Hope the answers and explanations helped!
