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Name:___________________________________ 90642 Calculate confidence intervals for population parameters. Credits: Three Answer ALL questions in the spaces provided in this booklet. Show ALL working for ALL questions. For Assessor’s use only Achievement Criteria Achievement Calculate confidence intervals for population parameters. Achievement with Merit Demonstrate an understanding of confidence intervals. Overall level of Performance Achievement with Excellence Demonstrate an understanding of the theory behind confidence intervals. Page 2 of 7 You are advised to spend 40 minutes answering the questions in this booklet. Show ALL working Question One A cell phone company is launching a new phone called the zPhone. (a) A random sample of 400 current customers found that 30% are planning on buying the zPhone. Calculate a 95% confidence interval for the proportion of all the company’s current customers who are planning on buying the zPhone. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ (b) Before the launch of the phone, a total of 754 pre-orders had been made for the zPhone by current customers of the cell phone company. Another random sample of 160 current customers made just before the launch of the phone found that 76 had pre-ordered the zPhone. Use these results to calculate a 95% confidence interval for the total number of current customers of the cell phone company. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ Page 3 of 7 (c) One week before launching the new zPhone, the cell phone company takes another survey of their current customers to find out how many are planning on buying the zPhone. They randomly sampled 100 current customers and found that the standard error for the estimate was twice the size of the standard error determined in (a). They also found that the proportion of current customers who are planning on buying the zPhone had increased, compared to the sample results in (a). Use these results to determine how many of the 100 customers sampled are planning on buying the zPhone. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ Page 4 of 7 Question Two The cell phone company offers two plans for customers who mostly use their phones to send texts: a super plan and a texter plan. The super plan costs $40 and includes 400 free texts per month, where as the texter plan costs $30 and includes 200 free texts per month. They used data from February this year to produce the table below: Sample mean number of texts sent Sample standard deviation Sample size (number of customers) Customers on super plan Customers on texter plan 386.4 267.4 84.8 69.6 204 259 Let μ1 be the mean number of texts sent for all customers on the super plan over one month and μ2 be the mean number of texts sent for all customers on the texter plan over one month. (a) Calculate a 99% confidence interval for μ1 - μ2. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ (b) Does this confidence interval suggest that customers on the super plan send on average 200 more texts per month than customers on the texter plan (the difference in the allocated free texts for the two plans)? You must justify your answer with statistical reasoning. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ Page 5 of 7 Data collected over the previous year indicated that the mean number of texts sent per month for all customers on the super plan is 400.8. (c) Using the standard deviation of 84.8, calculate the probability of obtaining a sample mean as small as 386.4 for a sample of 204 customers, and interpret what this suggests about this particular sample of customers and the number of texts they send. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ Page 6 of 7 Question Three The manufacturers of the zPhone claim that the battery for the phone will last for at least 5 hours of talking it will need recharging. This is called the talk time of a phone. (a) A random sample of 50 zPhones was taken. For the sample of phones, the mean talk time was 5.4 hours, and the standard deviation for the talk times was 0.4 hours. Calculate a 96% confidence interval for the mean talk time of all zPhones. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ (b) The manufacturers want to estimate the mean talk time for all zPhones to within 6 minutes of the true value with 96% confidence. Assuming a standard deviation for talk times of 0.4 hours, calculate the minimum number of zPhones they should sample. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ Page 7 of 7 (c) The weights of completely packaged zPhones (including the box, manuals and packing materials) have a mean weight of 410 g and a standard deviation of 4 g. zPhones are shipped in cartons, where each carton contains 100 zPhones. Calculate a 96% confidence interval for T, the true total weight of a carton of 100 zPhones. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________