Hypothesis Testing (2) 1. (a) A manufacturer of rockets estimates that, on average, 1 in 75 of the rockets fail to burn properly.Using this estimate, and a Poisson distribution, find an approximate value for the probability that, out of 225 randomly chosen rockets, at most 22l burn properly. State fully what distribution should be used to obtain the exact value of this probability. (b) 'Brilliant' fireworks are intended to burn for 40 seconds. A random sample of 50 'Brilliant' fireworks is taken. Each firework in the sample is ignited and the burning time, x seconds, is measured. The results are summarised by ∑ ,∑( – . Test, at the 5% level of significance, whether ) the mean burning time bf 'Brilliant' fireworks differs from 40 seconds. State, with a reason, whether, in using the above test, it is necessary to assume that the burning times of 'Brilliant' fireworks have a normal distribution. 2. A random sample of 90 batteries, used in a particular model of mobile phone, is tested and the 'standby- time', hours, is measured. The results are summarised by ∑ and∑ . Test, at the 1% significance level, whether the mean standby-time is less than 36.0 h. 3. The masses of the contents of tins of fish, of a certain brand, may be assumed to be normally distributed with unknown mean and unknown standard deviation. A random sample of 15 tins is selected from a large batch, and the contents weighed. The results are to be used to test, at the 5% significance level, the hypothesis that the mean mass of the batch differs from 50 grams. (i) State, giving your reasons, whether a normal test (z-test) or a t-test should be used. (ii) Suppose a trainee statistician uses the wrong one of the two tests in part (i). Explaining your answer, state whether it is possible that the trainee's test would lead to acceptance of the null hypothesis when the correct test would have led to its rejection. (iii) The masses, grams, of the contents of the 15 tins are summarised by ∑ Carry out the correct test. 4. The speeds of 120 randomly selected cars are measured as they pass a camera on a motorway. Denoting the speed by x km per hour, the results are summarised by ∑ , ∑( – ) Giving your answers correct to 2 places of decimals, find the unbiased estimates of the population mean and variance. 5. The random variabte X has the distribution N(1, 20). (i) Given that , find a. (ii) A random sample of n observations of X is taken. Given that the probability that the sample mean exceeds 1.5 is at most 0.01, find the set of possible values of n. 6. The mass, .r kg, of the contents of each packet in a random sample of 80 cereal packets is measured, and the results are summarised by ∑ ∑ , Test, at the 4% significance level, whether the population mean mass of the contents is less than 1.10 kg. , ∑ with a sample standard deviation of 5. to test whether the mean of X is 4. provide a 95% confidence interval for the mean assembly time. the sample standard deviation is 91. at the 5% level of significance. 9. A random sample of 10 observations of a normal variable X has mean x. . …. From the small data set in 2006. Provide a 90% confidence interval (to 2dp) for the average difference (right . In a very large study in the March quarter of the previous year it was found that the standard deviation of the usage was 81kWh. In a study to investigate this. the hypotheses are as follows. 8. 10. State your null and alternative hypotheses clearly. Null hypothesis: the population mean mass of the contents is equal to kg Alternative hypothesis: the population mean mass of the contents is not equal to kg Given that the null hypothesis is rejected in favour of the alternative hypothesis. find the set of possible values of . 7. and the number of kilowatt-hours (kWh) was recorded for each household in the sample for the March quarter of 2006. where ∑ ̅ ̅ Carry out a 2-tail test.left). and elementary calculations gave∑ and ∑ . The differences (right . provide an expression for calculating a 99% confidence interval for the mean usage in the March quarter of 2006.left) in diopters were . Assuming normality of assembly times. A random sample of 30 households was selected as part of a study on electricity usage. An industrial designer wants to determine the average amount of time it takes an adult to assemble an “easy to assemble” toy. The average usage was found to be 375kWh. Assuming the standard deviation is unchanged and that the usage is normally distributed. using the same data and also at the 47o significance level.73 minutes.92 minutes. . refraction was measured on the left and right eye of 17 patients.5kWh. Assuming that the usage is normally distributed. provide an expression for calculating a 99% confidence interval for the mean usage in the March quarter of 2006. 11. A sample of 16 times yielded an average time of 19. A topic of interest in ophthalmology is whether or not spherical refraction differs between the left and right eye on average. It is believed that the standard deviation may have changed from the previous year.Hypothesis Testing (2) In another test.58. Questions 8 and 9 use the same information. Provide an approximate 95% confidence interval for the proportion vaccinated in that suburb. the student union decides to interview a random sample of full time students. 14. if it important that the interval be no longer than 1cm? You may assume that the population is normal with variance 9cm. In exploring possible sites for a convenience store in a large neighbourhood. 15. 13. To obtain an estimate of the proportion of „full time‟ university students who have a part time job in excess of 20 hours per week. A random sample of 100 preschool children in Camperdown revealed that only 60 had been vaccinated.1. construct a 95% confidence interval for the average sale price.1 of the true proportion. would a random sample of size n=100 from the council records be sufficient for a 95% confidence interval of this precision? 16. They want the length of their 95% confidence interval to be no greater than 0.Hypothesis Testing (2) 12. . Provide a conservative 90% confidence interval for the proportion vaccinated in that suburb. What is the smallest sample size required to provide a 95% confidence interval for a mean. The recommended retail price of a brand of designer jeans is $150. Questions 13 and 14 use the same information. The price of the jeans in a sample of 16 retailers is on average $141 with a sample standard deviation of 4. n should be taken? 17. If this is a „random‟ sample and the prices can be assumed to be normally distributed. the retail chain wants to know the proportion of ratepayers in favour of the proposal. What size sample. If the estimate is required to be within 0.
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