keep an eye on the production process

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Question 1 Battery lifetime A firm produces batteries that have a lifetime which is normally distributed with a mean of 360 minutes and a standard deviation of 30 minutes. The firm needs to keep an eye on the production process to ensure that everything is working properly and that batteries are not being produced that do not meet the advertised standard. This is done by calculating the mean of the sample. To do this they regularly select a sample of 36 batteries in order to test the process. (a) Describe the sampling distribution of the sample mean lifetime of batteries (b) State a range within which you would expect the middle 80% of the sample means to lie. (c) If the process were working correctly, what is the probability that a sample would produce a mean of less than 350 minutes? (d) Based on your answer to part (c), what would you conclude about the process if the sample produced a mean life of batteries of 350 minutes? (e) What is the probability that two samples in a row would have a mean life time of less than 350 minutes? Question 2 Production Rates A manufacturing company wishes to estimate the number of items that its workforce can produce on average each hour now that they have a new machine. The factory examined the records for a random sample of 8 hours over the past month. The hourly production rates for these 8 hours were: 142 175 162 158 190 154 160 185 (a) Calculate by hand and interpret the 99 percent confidence interval for the average number of items produced per hour. (b) What assumption did you make in order to answer part (a)? Question 3 Customer Satisfaction (a) An insurance company wants to estimate the proportion of people unsatisfied with their new telephone help service. A survey of 200 callers revealed 45 were unsatisfied with the service. Construct by hand a 95% confidence interval for the proportion of unsatisfied customers. (b) If the company wanted to estimate the sample proportion to within 2% of the population proportion, what sample size would be necessary assuming no prior estimate of the proportion is known? Question 4 Production Rates Continued A manufacturing company wishes to find out whether the productivity of its workforce has increased now that they have a new machine. The factory examined the records for a random sample of 8 hours over the past month. The hourly production rates for these 8 hours were: 142 175 162 158 190 154 160 185 Suppose the production rate before the use of the new machine is 150 units per hour. Test at 5% if the new machine has increased workers’ productivity on average using the critical value approach. Question 5 Take the Pepsi Challenge “Take the Pepsi Challenge” was a marketing campaign used recently by the Pepsi-Cola Company. Coca-Cola drinkers participated in a blind taste test where they were asked to taste unmarked cups of Pepsi and Coke and were asked to select their favourite. In a Pepsi commercial, an announcer stated that “more than half the Coke drinkers prefer the taste of Pepsi”. Suppose 150 Coke drinkers took the Pepsi Challenge and 83 preferred the taste of Pepsi. Test this claim at the 5 percent level of significance using the p-value approach.

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