Statistics IV Inferential Statistics Solutions Revision Section A

Statistics IV – Inferential Statistics Solutions: Revision – Section A CHAPTER 25

25 1. (i) Revision and Exam Style Questions: Section A The time, in hours, that each student spent sleeping on a school night was recorded for 550 secondary school students. The distribution of these times was found to be approximately normal with a mean of 7∙ 6 hours and a standard deviation of 0∙ 5 hours. Draw the normal distribution curve to show this data.

25 1. (ii) Revision and Exam Style Questions: Section A The time, in hours, that each student spent sleeping on a school night was recorded for 550 secondary school students. The distribution of these times was found to be approximately normal with a mean of 7∙ 6 hours and a standard deviation of 0∙ 5 hours. What amount of time did 95% of these students spend sleeping? 95% of the students are between μ − 2σ → μ + 2σ = 6· 6 → 8· 6 hrs

25 1. (iii) Revision and Exam Style Questions: Section A The time, in hours, that each student spent sleeping on a school night was recorded for 550 secondary school students. The distribution of these times was found to be approximately normal with a mean of 7∙ 6 hours and a standard deviation of 0∙ 5 hours. What percentage of students spent between 7∙ 1 and 8∙ 1 hours sleeping? 7· 1 → 8· 1 hrs =μ−σ→μ+σ 68% =

25 2. Revision and Exam Style Questions: Section A A large group of students have a mean weight of 68 kg, with a standard deviation of 3 kg. Use the empirical rule to find the weight interval that contains 95% of the students. 95% are within: μ − 2σ → μ + 2σ 68 − 2(3) → 68 + 2(3) 68 − 6 → 68 + 6 62 kg → 74 kg

25 Revision and Exam Style Questions: Section A 3. Deirdre claims that the weather forecasts on the local radio are no better than rolling a fair die. She predicts rain if the result is odd and no rain if the result is even. She records the weather for 30 days and finds that the forecast is correct on 20 of the days. (i) State the null and alternative hypotheses. H 0: The percentage of times her forecast is correct is 50% H 1: The percentage of times her forecast is correct is not 50%

25 Revision and Exam Style Questions: Section A 3. Deirdre claims that the weather forecasts on the local radio are no better than rolling a fair die. She predicts rain if the result is odd and no rain if the result is even. She records the weather for 30 days and finds that the forecast is correct on 20 of the days. (ii) Construct the confidence interval.

25 Revision and Exam Style Questions: Section A 3. Deirdre claims that the weather forecasts on the local radio are no better than rolling a fair die. She predicts rain if the result is odd and no rain if the result is even. She records the weather for 30 days and finds that the forecast is correct on 20 of the days. (ii) Construct the confidence interval.

25 Revision and Exam Style Questions: Section A 3. Deirdre claims that the weather forecasts on the local radio are no better than rolling a fair die. She predicts rain if the result is odd and no rain if the result is even. She records the weather for 30 days and finds that the forecast is correct on 20 of the days. (ii) Construct the confidence interval. > 49· 0 p < 0· 85

25 3. (iii) Revision and Exam Style Questions: Section A Deirdre claims that the weather forecasts on the local radio are no better than rolling a fair die. She predicts rain if the result is odd and no rain if the result is even. She records the weather for 30 days and finds that the forecast is correct on 20 of the days. Is Deirdre’s claim correct? We fail to reject the null hypothesis as 0· 5 falls within the interval. Therefore Deirdre’s claim is true.

25 4. (i) Revision and Exam Style Questions: Section A A candidate for election claims that 45% of voters will vote for him. His opponent wants to test this claim. She takes a random sample of 120 voters and finds that 48 people said they would vote for him. State the null and alternative hypotheses. H 0: The percentage of voters that will vote for the candidate is 45%. H 1: The percentage of voters that will vote for the candidate is not 45%.

25 4. (ii) Revision and Exam Style Questions: Section A A candidate for election claims that 45% of voters will vote for him. His opponent wants to test this claim. She takes a random sample of 120 voters and finds that 48 people said they would vote for him. Calculate the confidence interval. > 09· 0 − 4· 0 p < 0· 4 + 0· 09 > 31· 0 p < 0· 49

25 4. (iii) Revision and Exam Style Questions: Section A A candidate for election claims that 45% of voters will vote for him. His opponent wants to test this claim. She takes a random sample of 120 voters and finds that 48 people said they would vote for him. Is the candidate’s claim upheld? We fail to reject the null hypothesis as 0· 45 falls within the confidence interval. Therefore, the candidates claim is upheld.

25 5. (i) Revision and Exam Style Questions: Section A An estate agent claims to have an 80% success rate for selling a house within two months. A client wants to test this hypothesis. He looks at the last 60 houses they have sold and finds that 50 of them had been sold within two months. State the null and alternative hypotheses for this test. H 0: The percentage of houses sold within 2 months is 80%. H 1: The percentage of houses sold within 2 months is not 80%.

25 5. (ii) Revision and Exam Style Questions: Section A An estate agent claims to have an 80% success rate for selling a house within two months. A client wants to test this hypothesis. He looks at the last 60 houses they have sold and finds that 50 of them had been sold within two months. Set up the confidence interval for this test. > 13· 0 − 83· 0 p < 0· 83 + 0· 13 > 7· 0 p < 0· 96

25 5. (iii) Revision and Exam Style Questions: Section A An estate agent claims to have an 80% success rate for selling a house within two months. A client wants to test this hypothesis. He looks at the last 60 houses they have sold and found that 50 of them had been sold within two months. Is the estate agent’s claim correct? We fail to reject the null hypothesis as 0· 8 falls within the confidence interval. Therefore the estate agents claim is correct.

25 Revision and Exam Style Questions: Section A 6. Adult IQ scores have a normal distribution with mean of 100 and a standard deviation of 15. (i) Use the empirical rule to find the percentage of adults with scores between 70 and 130. μ = 100 σ = 15 30 = 70 − 100 : 70 (15)2 = 70 = μ − 2σ 130 : 130 − 100 = 30 = 2(15) = 130μ + 2σ 70 → 130 = μ − 2σ → μ + 2σ = 95%

25 6. (ii) Revision and Exam Style Questions: Section A Adult IQ scores have a normal distribution with mean of 100 and a standard deviation of 15. If 250 adults are randomly selected, about how many of them have an IQ between 85 and 130? 85: 100 − 85 = 15 = 85μ – σ = 130μ + 2σ = 130 → 85 μ −σ → μ + 2σ 68% + 13· 5% = 5%· 81 = 81· 5% of 250 = 203· 75 Since it is not possible to have 0· 7 5 of a person, the number of adults that have an IQ between 85 and 130 is 203.

25 7. (i) Revision and Exam Style Questions: Section A A machine fills packets with roast peanuts. The line manager takes a sample of 10 packets every hour. The mean weights of the samples are normally distributed with a mean of 106 g and a standard deviation of 2 g. Write down the percentage of the samples that are likely to have a sample mean of less than 100 g. μ = 106 g σ=2 g 100: 106 − 100 = 6 = 3(2) 100 = μ − 3σ 15% < · 0μ − 3σ 0· 15% of the samples have a weight < 100 g

Revision and Exam Style Questions: Section A 25 7. (ii) A machine fills packets with roast peanuts. The line manager takes a sample of 10 packets every hour. The mean weights of the samples are normally distributed with a mean of 106 g and a standard deviation of 2 g. Write down the percentage of the samples that are likely to have a sample mean of more than 110 g. = 106 + 2(2) = 110μ + 2σ 35% + 0· 15% > μ + 2σ· 2 2· 5% of the samples have a weight > 110 g