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

3. A single observation \(x\) is to be taken from \(X \sim \mathrm {~B} ( 12 , p )\) This observation is used to test \(\mathrm { H } _ { 0 } : p = 0.45\) against \(\mathrm { H } _ { 1 } : p > 0.45\)

1. Using a \(5 \%\) level of significance, find the critical region for this test.
2. State the actual significance level of this test. The value of the observation is found to be 9
3. State the conclusion that can be made based on this observation.
4. State whether or not this conclusion would change if the same test was carried out at the
   1. 10\% level of significance,
   2. \(1 \%\) level of significance.

A mobile phone company claims that each year \(5 \%\) of its customers have their mobile phone stolen. An insurance company claims this percentage is higher. A random sample of 30 of the mobile phone company's customers is taken and 4 of them have had their mobile phone stolen during the last year.

1. Test the insurance company's claim at the \(10 \%\) level of significance. State your hypotheses clearly. A new random sample of 90 customers is taken. A test is carried out using these 90 customers, to see if the percentage of customers who have had a mobile phone stolen in the last year is more than 5\%
2. Using a suitable approximation and a \(10 \%\) level of significance, find the critical region for this test.
   

6.

1. Define the critical region of a test statistic. A discrete random variable \(X\) has a Binomial distribution \(\mathrm { B } ( 30 , p )\). A single observation is used to test \(\mathrm { H } _ { 0 } : p = 0.3\) against \(\mathrm { H } _ { 1 } : p \neq 0.3\)
2. Using a \(1 \%\) level of significance find the critical region of this test. You should state the probability of rejection in each tail which should be as close as possible to 0.005
3. Write down the actual significance level of the test. The value of the observation was found to be 15 .
4. Comment on this finding in light of your critical region.
   

7. (a) Explain briefly what you understand by

1. a critical region of a test statistic,
2. the level of significance of a hypothesis test.
   (b) An estate agent has been selling houses at a rate of 8 per month. She believes that the rate of sales will decrease in the next month.
3. Using a \(5 \%\) level of significance, find the critical region for a one tailed test of the hypothesis that the rate of sales will decrease from 8 per month.
4. Write down the actual significance level of the test in part (b)(i). The estate agent is surprised to find that she actually sold 13 houses in the next month. She now claims that this is evidence of an increase in the rate of sales per month.
   (c) Test the estate agent's claim at the \(5 \%\) level of significance. State your hypotheses clearly.

6.

1. Explain what you understand by a hypothesis.
2. Explain what you understand by a critical region. Mrs George claims that 45\% of voters would vote for her.
   In an opinion poll of 20 randomly selected voters it was found that 5 would vote for her.
3. Test at the \(5 \%\) level of significance whether or not the opinion poll provides evidence to support Mrs George's claim. In a second opinion poll of \(n\) randomly selected people it was found that no one would vote for Mrs George.
4. Using a \(1 \%\) level of significance, find the smallest value of \(n\) for which the hypothesis \(\mathrm { H } _ { 0 } : p = 0.45\) will be rejected in favour of \(\mathrm { H } _ { 1 } : p < 0.45\)

7. A drugs company claims that \(75 \%\) of patients suffering from depression recover when treated with a new drug. A random sample of 10 patients with depression is taken from a doctor's records.

1. Write down a suitable distribution to model the number of patients in this sample who recover when treated with the new drug. Given that the claim is correct,
2. find the probability that the treatment will be successful for exactly 6 patients. The doctor believes that the claim is incorrect and the percentage who will recover is lower. From her records she took a random sample of 20 patients who had been treated with the new drug. She found that 13 had recovered.
3. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, the doctor's belief.
4. From a sample of size 20, find the greatest number of patients who need to recover for the test in part (c) to be significant at the \(1 \%\) level.

3. The manager of a gym claimed that the mean age of its customers is 30 years. A random sample of 75 customers is taken and their ages have a mean of 28.2 years and a standard deviation, \(s\), of 8.5 years.

1. Stating your hypotheses clearly and using a 10\% level of significance, test whether or not the manager's claim is supported by the data.
2. Explain the relevance of the Central Limit Theorem to your calculation in part (a).
3. State an additional assumption needed to carry out the test in part (a).

4. The number of emergency plumbing calls received per day by a local council was recorded over a period of 80 days. The results are summarised in the table below.

1. Show that the mean number of emergency plumbing calls received per day is 3.5 A council officer suggests that a Poisson distribution can be used to model the number of emergency plumbing calls received per day. He uses the mean from the sample above and calculates the expected frequencies shown in the table below.

   \(\boldsymbol { x }\)	0	1	2	3	4	5	6	7	8 or more
   Expected frequency	2.42	8.46	14.80	\(r\)	15.10	10.57	6.17	3.08	\(s\)

2. Calculate the value of \(r\) and the value of \(s\), giving your answers correct to 2 decimal places.
3. Test, at the \(5 \%\) level of significance, whether or not the Poisson distribution is a suitable model for the number of emergency plumbing calls received per day. State your hypotheses clearly.

5. A dance studio has 800 dancers of which \begin{displayquote} 452 are beginners
251 are intermediates
97 are professionals

1. Explain in detail how a stratified sample of size 50 could be taken.
2. State an advantage of stratified sampling rather than simple random sampling in this situation. \end{displayquote} Independent random samples of 80 beginners and 60 intermediates are chosen. Each of these dancers is given an assessment score, \(x\), based on the quality of their dancing. The results are summarised in the table below.

   	\(\bar { x }\)	\(s ^ { 2 }\)	\(n\)
   Beginners	31.7	57.3	80
   Intermediates	36.9	38.1	60

   The studio manager believes that the mean score of intermediates is more than 3 points greater than the mean score of beginners.
3. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test whether or not these data support the studio manager's belief.

3. Star Farm produces duck eggs. Xander takes a random sample of 20 duck eggs from Star Farm and their widths, \(x \mathrm {~cm}\), are recorded. Xander's results are summarised as follows. $$\sum x = 92.0 \quad \sum x ^ { 2 } = 433.4974$$

1. Calculate unbiased estimates of the mean and the variance of the width of duck eggs produced by Star Farm. Yinka takes an independent random sample of 30 duck eggs from Star Farm and their widths, \(y \mathrm {~cm}\), are recorded. Yinka's results are summarised as follows. $$\sum y = 142.5 \quad \sum y ^ { 2 } = 689.5078$$
2. Treating the combined sample of 50 duck eggs as a single sample, estimate the standard error of the mean.
   (5) Research shows that the population of duck egg widths is normally distributed with standard deviation 0.71 cm . The farmer claims that the mean width of duck eggs produced by Star Farm is greater than 4.5 cm .
3. Using your combined mean, test, at the \(5 \%\) level of significance, the farmer's claim. State your hypotheses clearly.

1. A doctor believes that the diet of her patients and their health are not independent.

She takes a random sample of 200 patients and records whether they are in good health or poor health and whether they have a good diet or a poor diet. The results are summarised in the table below. Stating your hypotheses clearly, test the doctor's belief using a \(5 \%\) level of significance. Show your working for your test statistic and state your critical value clearly.

A college runs academic and vocational courses. The college has 1680 academic students and 2520 vocational students.

1. Describe how a stratified sample of 70 students at the college could be taken. All students at the college take a basic skills test. A random sample of 50 academic students has a mean score of 57 and a variance of 60. An independent random sample of 80 vocational students has a mean score of 62 with a variance of 70
2. Stating your hypotheses clearly, test at the \(5 \%\) level of significance, whether or not the mean basic skills score for vocational students is greater than the mean basic skills score for academic students.
3. Explain the importance of the Central Limit Theorem to the test in part (b).
4. State an assumption that is required to carry out the test in part (b). All the academic students at the college take a basic skills course. Another random sample of 50 academic students and another independent random sample of 80 vocational students retake the basic skills test. The hypotheses used in part (b) are then tested again at the same level of significance. The value of the test statistic \(z\) is now 1.54
5. Comment on the mean basic skills scores of academic and vocational students after taking this course.
6. Considering the outcomes of the tests in part (b) and part (e), comment on the effectiveness of the basic skills course.

1. A researcher is looking into the effectiveness of a new medicine for the relief of symptoms. He collects random samples of 8 people who are taking the medicine from each of 50 different medical practices. The number of people who say that the medicine is a success, in each sample, is recorded. The results are summarised in the table below.

The researcher decides to model this data using a binomial distribution.

1. State two necessary assumptions that the researcher made in order to use this model.
2. Show that the mean number of successes per sample is 3.54 He decides to use this mean to calculate expected frequencies. The results are shown in the table below.

   Number of successes	0	1	2	3	4	5	6	7	8
   Expected frequency	0.47	2.96	8.23	13.07	\(f\)	8.23	3.27	0.74	\(g\)

3. Calculate the value of \(f\) and the value of \(g\). Give your answers to 2 decimal places.
4. Stating your hypotheses clearly, test at the \(10 \%\) level of significance, whether or not the binomial distribution is a suitable model for the number of successes in samples of 8 people.

1. An experiment is conducted to compare the heat retention of two brands of flasks, brand \(A\) and brand \(B\). Both brands of flask have a capacity of 750 ml .

In the experiment 750 ml of boiling water is poured into the flask, which is then sealed. Four hours later the temperature, in \({ } ^ { \circ } \mathrm { C }\), of the water in the flask is recorded. A random sample of 100 flasks from brand \(A\) gives the following summary statistics, where \(x\) is the temperature of the water in the flask after four hours. $$\sum x = 7690 \quad \sum ( x - \bar { x } ) ^ { 2 } = 669.24$$

1. Find unbiased estimates for the mean and variance of the temperature of the water, after four hours, for brand \(A\). A random sample of 80 flasks from brand \(B\) gives the following results, where \(y\) is the temperature of the water in the flask after four hours. $$\bar { y } = 75.9 \quad s _ { y } = 2.2$$
2. Test, at the \(1 \%\) significance level, whether there is a difference in the mean water temperature after four hours between brand \(A\) and brand \(B\). You should state your hypotheses, test statistic and critical value clearly.
3. Explain why it is reasonable to assume that \(\sigma ^ { 2 } = s ^ { 2 }\) in this situation.

3. A die is rolled 60 times, and results in 16 sixes.

1. Use a suitable approximation to test, at the \(5 \%\) significance level, whether the probability of scoring a six is \(\frac { 1 } { 6 }\) or not. State your hypotheses clearly.
2. Describe how you would change the test if you wished to investigate whether the probability of scoring a six is greater than \(\frac { 1 } { 6 }\). Carry out this modified test.

3. A coin is tossed 20 times, giving 16 heads.

1. Test at the \(1 \%\) significance level whether the coin is fair, stating your hypotheses clearly.
2. Find the critical region for the same test at the \(0.1 \%\) significance level.

3. The Driving Theory Test includes 30 questions which require one answer to be selected from four options.

1. Phil ticks answers at random. Find how many of the 30 he should expect to get right.
2. If he gets 15 correct, decide whether this is evidence that he has actually done some revision. Use a \(5 \%\) significance level. Another candidate, Sarah, has revised and has a 0.9 probability of getting each question right.
3. Determine the expected number of answers that Sarah will get right.
4. Find the probability that Sarah gets more than 25 correct answers out of 30.

3. The random variable \(X\) is modelled by a binomial distribution \(\mathrm { B } ( n , p )\), with \(n = 20\) and \(p\) unknown. It is suspected that \(p = 0 \cdot 4\).

1. Find the critical region for the test of \(\mathrm { H } _ { 0 } : p = 0.4\) against \(\mathrm { H } _ { 1 } : p \neq 0.4\), at the \(5 \%\) significance level.
2. Find the critical region if, instead, the alternative hypothesis is \(\mathrm { H } _ { 1 } : p < 0.4\).

5. A traffic analyst is interested in the number of heavy lorries passing a certain junction. He counts the numbers of lorries in 100 five-minute intervals, and gets the following results: Q. 5 continued on next page ... \section*{STATISTICS 2 (A) TEST PAPER 9 Page 2} continued ...

1. Show that the mean of \(X\) is 3 , and find the variance of \(X\).
2. Give two reasons for thinking that \(X\) can be modelled by a Poisson distribution. (2 marks) After a new landfill site has been established nearby, a member of an environmental group notices that 18 lorries pass the junction in a period of 15 minutes. The group claims that this is evidence that the mean number of lorries per five-minute interval has increased.
3. Test whether the group's claim is valid. Work at the \(5 \%\) significance level, and state your hypotheses clearly.
   

6. A teacher is monitoring attendance at lessons in her department. She believes that the number of students absent from each lesson follows a Poisson distribution and wished to test the null hypothesis that the mean is 2.5 against the alternative hypothesis that it is greater than 2.5 She visits one lesson and decides on a critical region of 6 or more students absent.

1. Find the significance level of this test.
2. State any assumptions made in carrying out this test and comment on their validity. The teacher decides to undertake a wider study by looking at a sample of all the lessons that have taken place in the department during the previous four weeks.
3. Suggest a suitable sampling frame. She finds that there have been 96 pupils absent from the 30 lessons in her sample.
4. Using a suitable approximation, test at the \(5 \%\) level of significance the null hypothesis that the mean is 2.5 students absent per lesson against the alternative hypothesis that it is greater than 2.5. You may assume that the number of absences follows a Poisson distribution.
   (6 marks)

2. A driving instructor keeps records of all the learners she has taught. In order to analyse her success rate she wishes to take a random sample of 120 of these learners.

1. Suggest a suitable sampling frame and identify the sampling units. She believes that only 1 in 20 of the people she teaches fail to pass their test in their first two attempts. She decides to use her sample to test whether or not the proportion is different from this.
2. Using a suitable approximation and stating clearly the hypotheses she should use, find the largest critical region for this test such that the probability in each "tail" is less than \(2.5 \%\).
3. State the significance level of this test.

5. Four coins are flipped together and the random variable \(H\) represents the number of heads obtained. Assuming that the coins are fair,

1. suggest with reasons a suitable distribution for modelling \(H\) and give the value of any parameters needed,
2. show that the probability of obtaining more heads than tails is \(\frac { 5 } { 16 }\). The four coins are flipped 5 times and more heads are obtained than tails 4 times.
3. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence of the probability of getting more heads than tails being more than \(\frac { 5 } { 16 }\). Given that the four coins are all biased such that the chance of each one showing a head is 50\% more than the chance of it showing a tail,
4. find the probability of obtaining more heads than tails when the four coins are flipped together.

2. (a) Define

1. a Type I error,
2. a Type II error. A manufacturer sells socks in boxes of 50 .
   The mean number of faulty socks per box is 7.5 . In order to reduce the number of faulty socks a new machine is tried. A box of socks made on the new machine was tested and the number of faulty socks was 2.
   (b) (i) Assuming that the number of faulty socks per box follows a binomial distribution derive a critical region needed to test whether or not there is evidence that the new machine has reduced the mean number of faulty socks per box. Use a \(5 \%\) significance level.
3. Stating your hypotheses clearly, carry out the test in part (i).
   (c) Find the probability of the Type I error for this test.
   (d) Given that the true mean number of faulty socks per box on the new machine is 5 , calculate the probability of a Type II error for this test.
   (e) Explain what would have been the effect of changing the significance level for the test in part (b) to \(2 \frac { 1 } { 2 } \%\).

4. The number of accidents that occur at a crossroads has a mean of 3 per month. In order to improve the flow of traffic the priority given to traffic is changed. Colin believes that since the change in priority the number of accidents has increased. He tests his belief by recording the number of accidents \(x\) in the month following the change. Colin sets up the hypotheses \(\mathrm { H } _ { 0 } : \lambda = 3\) and \(\mathrm { H } _ { 1 } : \lambda > 3\), where \(\lambda\) is the mean number of accidents per month, and rejects the null hypothesis if \(x > 4\).

1. Find the size of the test. The table gives the values of the power function of the test to two decimal places.

   \(\lambda\)	4	5	6	7
   Power	\(r\)	0.56	\(s\)	0.83

2. Calculate the value of \(r\) and the value of \(s\).
3. Comment on the suitability of the test when \(\lambda = 4\).