Hypothesis test then Type II error probability

6 The masses of cereal boxes filled by a certain machine have mean 510 grams. An adjustment is made to the machine and an inspector wishes to test whether the mean mass of cereal boxes filled by the machine has decreased. After the adjustment is made, he chooses a random sample of 120 cereal boxes. The mean mass of these boxes is found to be 508 grams. Assume that the standard deviation of the masses is 10 grams.

1. Test at the \(2.5 \%\) significance level whether the mean mass of cereal boxes filled by the machine has decreased.
   Later the inspector carries out a similar test at the \(2.5 \%\) significance level, using the same hypotheses and another 120 randomly chosen cereal boxes.
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2. Given that the mean mass is now actually 506 grams, find the probability of a Type II error. [5]

7 The masses, in grams, of apples from a certain farm have mean \(\mu\) and standard deviation 5.2. The farmer says that the value of \(\mu\) is 64.6. A quality control inspector claims that the value of \(\mu\) is actually less than 64.6. In order to test his claim he chooses a random sample of 100 apples from the farm.

1. The mean mass of the 100 apples is found to be 63.5 g . Carry out the test at the \(2.5 \%\) significance level.
2. Later another test of the same hypotheses at the \(2.5 \%\) significance level, with another random sample of 100 apples from the same farm, is carried out. Given that the value of \(\mu\) is in fact 62.7 , calculate the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 In the past Laxmi's time, in minutes, for her journey to college had mean 32.5 and standard deviation 3.1. After a change in her route, Laxmi wishes to test whether the mean time has decreased. She notes her journey times for a random sample of 50 journeys and she finds that the sample mean is 31.8 minutes. You should assume that the standard deviation is unchanged.

1. Carry out a hypothesis test, at the \(8 \%\) significance level, of whether Laxmi's mean journey time has decreased.
   Later Laxmi carries out a similar test with the same hypotheses, at the \(8 \%\) significance level, using another random sample of size 50 .
2. Given that the population mean is now 31.5, find the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 A biologist wishes to test whether the mean concentration \(\mu\), in suitable units, of a certain pollutant in a river is below the permitted level of 0.5 . She measures the concentration, \(x\), of the pollutant at 50 randomly chosen locations in the river. The results are summarised below. $$n = 50 \quad \Sigma x = 23.0 \quad \Sigma x ^ { 2 } = 13.02$$

1. Carry out a test at the \(5 \%\) significance level of the null hypothesis \(\mu = 0.5\) against the alternative hypothesis \(\mu < 0.5\).
   Later, a similar test is carried out at the \(5 \%\) significance level using another sample of size 50 and the same hypotheses as before. You should assume that the standard deviation is unchanged.
2. Given that, in fact, the value of \(\mu\) is 0.4 , find the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8 In order to test the effect of a drug, a researcher monitors the concentration, \(X\), of a certain protein in the blood stream of patients. For patients who are not taking the drug the mean value of \(X\) is 0.185 . A random sample of 150 patients taking the drug was selected and the values of \(X\) were found. The results are summarised below. $$n = 150 \quad \Sigma x = 27.0 \quad \Sigma x ^ { 2 } = 5.01$$ The researcher wishes to test at the \(1 \%\) significance level whether the mean concentration of the protein in the blood stream of patients taking the drug is less than 0.185 .

1. Carry out the test.
2. Given that, in fact, the mean concentration for patients taking the drug is 0.175 , find the probability of a Type II error occurring in the test.

7 A mill owner claims that the mean mass of sacks of flour produced at his mill is 51 kg . A quality control officer suspects that the mean mass is actually less than 51 kg . In order to test the owner's claim she finds the mass, \(x \mathrm {~kg}\), of each of a random sample of 150 sacks and her results are summarised as follows. $$n = 150 \quad \Sigma x = 7480 \quad \Sigma x ^ { 2 } = 380000$$

1. Carry out the test at the \(2.5 \%\) significance level.
   You may now assume that the population standard deviation of the masses of sacks of flour is 6.856 kg . The quality control officer weighs another random sample of 150 sacks and carries out another test at the 2.5\% significance level.
2. Given that the population mean mass is 49 kg , find the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 Bob is a self-employed builder. In the past his weekly income had mean \(\\) 546\( and standard deviation \)\\( 120\). Following a change in Bob's working pattern, his mean weekly income for 40 randomly chosen weeks was \(\\) 581\(. You should assume that the standard deviation remains unchanged at \)\\( 120\).

1. Test at the \(2.5 \%\) significance level whether Bob's mean weekly income has increased.
   Bob finds his mean weekly income for another random sample of 40 weeks and carries out a similar test at the \(2.5 \%\) significance level.
2. Given that Bob's mean weekly income is now in fact \(\\) 595$, find the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 Parcels arriving at a certain office have weights \(W \mathrm {~kg}\), where the random variable \(W\) has mean \(\mu\) and standard deviation 0.2 . The value of \(\mu\) used to be 2.60 , but there is a suspicion that this may no longer be true. In order to test at the 5\% significance level whether the value of \(\mu\) has increased, a random sample of 75 parcels is chosen. You may assume that the standard deviation of \(W\) is unchanged.

1. The mean weight of the 75 parcels is found to be 2.64 kg . Carry out the test.
2. Later another test of the same hypotheses at the \(5 \%\) significance level, with another random sample of 75 parcels, is carried out. Given that the value of \(\mu\) is now 2.68 , calculate the probability of a Type II error.