The supplier claims that the mean mass of boxes of cereal is 253 g . A quality control officer suspects that the mean mass is actually more than 253 g . In order to test this claim, he weighs a random sample of 100 boxes of cereal and finds that the total mass is 25360 g .

[ 6 ]

(i)

Given that the population standard deviation of the masses is 3.5 g, test at the 5% significance level whether the population mean mass is more than 253 g.

An employee says, 'This test is invalid because it uses the normal distribution, but we do not know whether the masses of the boxes are normally distributed.'

[ 5 ]

(ii)

Explain briefly whether this statement is true or not.

[ 1 ]

In this question you should not use an approximating distribution.
At an election in Menham last year, 24 % of voters supported the Today Party. A student wishes to test whether support for the Today Party has decreased since last year. He chooses a random sample of 25 voters in Menham and finds that exactly 2 of them say that they support the Today Party.

Test at the 5\% significance level whether support for the Today Party has decreased.

[Maximum number: 5]

A teacher models the numbers of girls and boys who arrive late for her class on any day by the independent random variables $G \sim \operatorname{Po}(0.10)$ and $B \sim \operatorname{Po}(0.15)$ respectively.

(a)

Test the teacher's claim at the 5% significance level.

[ 5 ]

[Maximum number: 6]

The numbers of green sweets in 200 randomly chosen packets of Frutos are summarised in the table.

Number of green sweets	0	1	2	3	>3
Number of packets	32	50	97	21	0

(a)

The manufacturers of Frutos claim that the mean number of green sweets in a packet is 1.65 .
Anji believes that the true value of the mean, $\mu$ , is less than 1.65 . She uses the results from the 200 randomly chosen packets to test the manufacturers' claim.

State suitable null and alternative hypotheses for the test.

[ 1 ]

(b)

Show that the result of Anji's test is significant at the 5 % level but not at the 1 % level.

[ 4 ]

(c)

It is given that Anji made a Type I error.

Explain how this shows that the significance level that Anji used in her test was not 1 %.

[ 1 ]

[Maximum number: 7]

Every July, as part of a research project, Rita collects data about sightings of a particular kind of bird. Each day in July she notes whether she sees this kind of bird or not, and she records the number X of days on which she sees it. She models the distribution of X by B(31, p), where p is the probability of seeing this kind of bird on a randomly chosen day in July.

Data from previous years suggests that p=0.3, but in 2022 Rita suspected that the value of p had been reduced. She decided to carry out a hypothesis test.

In July 2022, she saw this kind of bird on 4 days.

(a)

Use the binomial distribution to test at the 5% significance level whether Rita's suspicion is justified.

In July 2023, she noted the value of X and carried out another test at the 5% significance level using the same hypotheses.

[ 5 ]

(b)

Calculate the probability of a Type I error.

Rita models the number of sightings, Y, per year of a different, very rare, kind of bird by the distribution B(365,0.01).

[ 2 ]

[Maximum number: 6]

The heights, in centimetres, of adult females in Litania have mean $\mu$ and standard deviation $\sigma$ . It is known that in 2004 the values of $\mu$ and $\sigma$ were 163.21 and 6.95 respectively. The government claims that the value of $\mu$ this year is greater than it was in 2004. In order to test this claim a researcher plans to carry out a hypothesis test at the 1% significance level. He records the heights of a random sample of 300 adult females in Litania this year and finds the value of the sample mean.

(a)

State the probability of a Type I error.

You should assume that the value of $\sigma$ after 2004 remains at 6.95 .

[ 1 ]

(b)

Given that the value of $\mu$ this year is actually 164.91, find the probability of a Type II error.

[ 5 ]

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Question 4

Question 4(b)

Question 4(b)(i)

Question 4(b)(ii)

Question 4

Question 5

Question 5(d)

Question 6

Question 6(b)

Question 6(c)

Question 6(d)

Question 7

Question 7(a)

Question 7(b)

Question 7

Question 7(a)

Question 7(b)