2.05c Significance levels: one-tail and two-tail

A company produces sweets of varying colours. The company claims that the proportion of blue sweets is 13·6%. A consumer believes that the true proportion is less than this. In order to test this belief, the consumer collects a random sample of 80 sweets.

1. State suitable hypotheses for the test. [1]
2. 1. Determine the critical region if the test is to be carried out at a significance level as close as possible to, but not exceeding, 5%.
   2. Given that there are 6 blue sweets in the sample of 80, complete the significance test. [5]
3. Suppose the proportion of blue sweets claimed by the company is correct. The consumer conducts the sampling and testing process on a further 20 occasions, using the sample size of 80 each time. What is the expected number of these occasions on which the consumer would reach the incorrect conclusion? [2]
4. Now suppose that the proportion of blue sweets is 7%. Find the probability of a Type II error. Interpret your answer in context. [3]

Dewi, a candidate in an election, believes that 45% of the electorate intend to vote for him. His agent, however, believes that the support for him is less than this. Given that \(p\) denotes the proportion of the electorate intending to vote for Dewi,

1. state hypotheses to be used to resolve this difference of opinion. [1]

They decide to question a random sample of 60 electors. They define the critical region to be \(X \leq 20\), where \(X\) denotes the number in the sample intending to vote for Dewi.

1. 1. Determine the significance level of this critical region.
   2. If in fact \(p\) is actually 0.35, calculate the probability of a Type II error.
   3. Explain in context the meaning of a Type II error.
   4. Explain briefly why this test is unsatisfactory. How could it be improved while keeping approximately the same significance level? [8]

Arlosh, Sarah and Desi are investigating the ratings given to six different films by two critics.

1. Arlosh calculates Spearman's rank correlation coefficient \(r_s\) for the critics' ratings. He calculates that \(\Sigma d^2 = 72\). Show that this value must be incorrect. [2]
2. Arlosh checks his working with Sarah, whose answer \(r_s = \frac{39}{35}\) is correct. Find the correct value of \(\Sigma d^2\). [2]
3. Carry out an appropriate two-tailed significance test of the value of \(r_s\) at the 5% significance level, stating your hypotheses clearly. [4]

Tiana is a quality controller in a clothes factory. She checks for four possible types of defects in shirts. Of the shirts with defects, the proportion of each type of defect is as shown in the table below. Tiana wants to investigate the proportion, \(p\), of defective shirts with a fabric defect. She wishes to test the hypotheses $$H_0 : p = 0.3$$ $$H_1 : p < 0.3$$ She takes a random sample of 60 shirts with a defect and finds that \(x\) of them have a fabric defect.

1. Using a 5% level of significance, find the critical region for \(x\). [5 marks]
2. In her sample she finds 13 shirts with a fabric defect. Complete the test stating her conclusion in context. [2 marks]

A firm claims that no more than 2\% of their packets of sugar are underweight. A market researcher believes that the actual proportion is greater than 2\%. In order to test the firm's claim, the researcher weighs a random sample of 600 packets and carries out a hypothesis test, at the 5\% significance level, using the null hypothesis \(p = 0.02\).

1. Given that the researcher's null hypothesis is correct, determine the probability that the researcher will conclude that the firm's claim is incorrect. [5]
2. The researcher finds that 18 out of the 600 packets are underweight. A colleague says "18 out of 600 is 3\%, so there is evidence that the actual proportion of underweight bags is greater than 2\%." Criticise this statement. [1]

It is known that, under standard conditions, 12% of light bulbs from a certain manufacturer have a defect. A quality improvement process has been implemented, and a random sample of 200 light bulbs produced after the improvements was selected. It was found that 15 of the 200 light bulbs were defective.

1. State one assumption needed in order to use a binomial model for the number of defective light bulbs in the sample. [1]
2. Test, at the 5% significance level, whether the proportion of defective light bulbs has decreased under the new process. [7]

A television company believes that the proportion of households that can receive Channel C is 0.35.

1. In a random sample of 14 households it is found that 2 can receive Channel C. Test, at the 2.5\% significance level, whether there is evidence that the proportion of households that can receive Channel C is less than 0.35. [7]
2. On another occasion the test is carried out again, with the same hypotheses and significance level as in part (i), but using a new sample, of size \(n\). It is found that no members of the sample can receive Channel C. Find the largest value of \(n\) for which the null hypothesis is not rejected. Show all relevant working. [4]

Research has shown that drug A is effective in 32% of patients with a certain disease. In a trial, drug B is given to a random sample of 1000 patients with the disease, and it is found that the drug is effective in 290 of these patients. Test at the 2.5% significance level whether there is evidence that drug B is effective in a lower proportion of patients than drug A. [7]

Laxmi wishes to test whether there is linear correlation between the mass and the height of adult males.

1. State, with a reason, whether Laxmi should use a 1-tail or a 2-tail test. [1]

Laxmi chooses a random sample of 40 adult males and calculates Pearson's product-moment correlation coefficient, \(r\). She finds that \(r = 0.2705\).

1. Use the table below to carry out the test at the 5% significance level. [5]

Critical values of Pearson's product-moment correlation coefficient.