A machine in the student lounge dispenses coffee. The average cup of coffee is supposed to contain 7.0 ounces. A random sample of seven cups of coffee from this machine show the average content to be 7.4 ounces with a standard deviation of 0.70 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

Answer: To determine if the coffee machine has slipped out of adjustment and is dispensing a different average amount of coffee from 7 ounces, you can use a hypothesis test.

The first step is to state the null hypothesis (H0) and the alternative hypothesis (H1): - H0: µ = 7.0 (The machine is correctly adjusted, and the true average is 7.0 ounces) - H1: µ ≠ 7.0 (The machine has slipped out of adjustment, and the true average is not 7.0 ounces)

For this situation, you would likely use a t-test since the sample size is small (n=7), and you presumably do not know the population standard deviation.

Next, you calculate the test statistic using the sample data: The formula for the t-statistic is: t = (x̄ - µ) / (s/√n) where: - x̄ = sample mean (7.4 ounces) - µ = hypothesized population mean (7.0 ounces) - s = sample standard deviation (0.70 ounce) - n = sample size (7)

Plugging in the given numbers: t = (7.4 - 7.0) / (0.70 / √7) t ≈ (0.4) / (0.2646) t ≈ 1.5119

Now you need to compare this test statistic to a critical value from the t-distribution table, which will depend on the chosen significance level (a common choice is 0.05 for a two-tailed test) and the degrees of freedom (n-1 = 6 in this case).

If the absolute value of the calculated t-value is greater than the critical t-value, then you reject the null hypothesis. If it is less, you do not have enough evidence to reject the null hypothesis.

Assuming a significance level of 0.05, the critical t-value for a two-tailed test with 6 degrees of freedom is approximately 2.447. Since the absolute value of our calculated t-value (1.5119) is less than 2.447, we do not reject the null hypothesis.

The conclusion would be that there is not sufficient evidence to say that the machine is dispensing a different average amount of coffee from 7 ounces.