ICE CREAM PROBLEM – IS THE MACHINE OUT OF ADJUSTMENT?

A machine fills 64-ounce cartons with ice cream. The manager thinks that the machine may need to be adjusted because it is putting into the cartons an amount of ice cream different than 64 ounces.

A random sample of 16 ice cream cartons was selected showing a sample mean of 64.21 ounces with a standard deviation s of 0.72 ounces.

We would like to explain why the x-bar of 64.21 is different than 64.

Is it different by chance, just because of the random selection we made? Or, is it significantly different; which may imply that the machine is out of adjustment?

Hypothesis testing

A hypothesis test was performed for ounces, producing the results: t = 1.146 and p = 0.270. The large p-value is an indication that a sample mean of 64.21 is very likely to happen when mu is 64 ounces. A sample mean of 64.21 is not statistically significant (at any alpha level). There is not good evidence that the machine is putting into the cartons on average, an amount of ice cream different than 64 ounces. Sample results do not support the manager’s claim that the machine is out of adjustment.

What does the p-value say?

If the population mean is 64, a sample mean of 64.21 or a more extreme one may be observed in 27 out of 100 cases (p = 0.27). The large probability says that such an x-bar is very likely to be observed when the population mean is 64. We may explain the difference between the x-bar of 64.21 and mu of 64 by chance variation.

Confidence interval

A 95% confidence interval estimate for mu was also constructed producing the interval (63.83, 64.59). With 95% confidence we can say that on average the machine puts between 63.83 ounces and 64.59 ounces of ice cream into the cartons. Since 64 is in the interval, it’s possible that the average amount being put into the cartons is 64 ounces which does not support the manager’s claim that the machine is out of adjustment.

What does the 95% confidence level mean?

  • 95% refers to the success rate of the method used to construct the interval.
  • 95% of the intervals constructed with this method will contain the true mean mu.
  • If we select 100 samples of 16 cartons, and around each sample mean we construct the corresponding 95% interval estimate, we expect that about 95 intervals will contain mu, and about 5 of them will not.

We are not sure whether our interval contains mu or not; we are just 95% confident that it does.

What IT DOES NOT mean?

  • FALSE:There is a 95% probability that this interval contains mu
  • REMEMBER THAT the probability that this interval contains mu is either 1 or 0

TESTING CENTER PROBLEM: TIME IT TAKES TO COMPLETE A BUSINESS STATISTIC TEST

The average time required to take the Business Statistics test (in minutes) is being questioned. The testing center that prepares the test advertises a mean time of 40 minutes and some professors are saying that, on average, students take more than 40 minutes to finish the test.

Results from 31 business statistics students show a mean of 43.5 minutes. Sigma is given as 8 minutes.

We would like to explain why the x-bar of 43.5 is higher than 40.

Is it higher by chance, just because of the random selection we made? Or, is it significantly higher; which may imply that on average students take more than 40 minutes to finish the test?

Hypothesis testing

A hypothesis test was performed for mu > 40 minutes, with results: z = 2.44 and p = 0.007. The results indicate that the sample mean of 43.5 minutes is significantly higher than 40 (statistically significant at the 5% significance level) and it provides good evidence that the mean time to finish the test is more than 40 minutes. The testing center should revise the test in order to fit the advertised 40-minute average.

What does the p-value say?

If the population mean is 40 minutes, the probability that a sample of 31 students have a mean test score of at least 43.5 minutes is 0.007. Such sample means can be observed in 7 out of 1000 cases; this unlikely result of 43.5 minutes gives good evidence that the mean time to finish the test is more than the advertised 40 minutes. (43.5 is significantly higher than 40)

Confidence interval

A 90% confidence interval estimate for mu was also constructed producing the results: (41.136, 45.864)

With 90% confidence we can say that the mean time to finish the test is between 41.136 minutes and 45.864 minutes; values that are all more than 40 minutes. These results also give evidence that, on average, students take more than 40 minutes to finish the test. Again, a revision of the test is recommended.

What does the 90% confidence level mean?

  • 90% refers to the success rate of the method used to construct the interval.
  • 90% of the intervals constructed with this method will contain the true mean mu.
  • If we select 100 samples of 31 students, and around each sample mean we construct the corresponding 90% interval estimate, we expect that about 90 intervals will contain mu, and about 10 of them will not.

We are not sure whether our interval contains mu or not; we are just 90% confident that it does.