2.05a Hypothesis testing language: null, alternative, p-value, significance

Ellie, a statistics student, read a newspaper article that stated that 20 per cent of students eat at least five portions of fruit and vegetables every day. Ellie suggests that the number of people who eat at least five portions of fruit and vegetables every day, in a sample of size \(n\), can be modelled by the binomial distribution B(\(n\), 0.20).

1. There are 10 students in Ellie's statistics class. Using the distributional model suggested by Ellie, find the probability that, of the students in her class:
   1. two or fewer eat at least five portions of fruit and vegetables every day; [1 mark]
   2. at least one but fewer than four eat at least five portions of fruit and vegetables every day; [2 marks]
2. Ellie's teacher, Declan, believes that more than 20 per cent of students eat at least five portions of fruit and vegetables every day. Declan asks the 25 students in his other statistics classes and 8 of them say that they eat at least five portions of fruit and vegetables every day.
   1. Name the sampling method used by Declan. [1 mark]
   2. Describe one weakness of this sampling method. [1 mark]
   3. Assuming that these 25 students may be considered to be a random sample, carry out a hypothesis test at the 5\% significance level to investigate whether Declan's belief is supported by this evidence. [6 marks]

Edward can correctly identify 20% of types of wild flower. He studies some books to see if he can improve how often he can correctly identify types of wild flower. He collects a random sample of 10 types of wild flower in order to test whether or not he has improved.

1. 1. Write suitable hypotheses for this test.
   2. State a suitable test statistic that he could use. [2]
2. Using a 5% level of significance, find the critical region for this test. [3]
3. State the probability of a Type I error for this test and explain what it means in this context. [2]
4. Edward correctly identifies 4 of the 10 types of wild flower he collected. What conclusion should Edward reach? [2]

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]

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]