15 A quality control department checks the lifetimes of batteries produced by a company.
The lifetimes, \(x\) minutes, for a random sample of 80 'Superstrength' batteries are shown in the table below.
| Lifetime | \(160 \leq x < 165\) | \(165 \leq x < 168\) | \(168 \leq x < 170\) | \(170 \leq x < 172\) | \(172 \leq x < 175\) | \(175 \leq x < 180\) |
| Frequency | 5 | 14 | 20 | 21 | 16 | 4 |
- Estimate the proportion of these batteries which have a lifetime of at least 174.0 minutes.
- Use the data in the table to estimate
- the sample mean,
- the sample standard deviation.
The data in the table on the previous page are represented in the following histogram, Fig 15:
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\caption{Fig. 15}
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A quality control manager models the data by a Normal distribution with the mean and standard deviation you calculated in part (b). - Comment briefly on whether the histogram supports this choice of model.
- Use this model to estimate the probability that a randomly selected battery will have a lifetime of more than 174.0 minutes.
- Compare your answer with your answer to part (a).
The company also manufactures 'Ultrapower' batteries, which are stated to have a mean lifetime of 210 minutes.
- A random sample of 8 Ultrapower batteries is selected. The mean lifetime of these batteries is 207.3 minutes.
Carry out a hypothesis test at the \(5 \%\) level to investigate whether the mean lifetime is as high as stated. You should use the following hypotheses \(\mathrm { H } _ { 0 } : \mu = 210 , \mathrm { H } _ { 1 } : \mu < 210\), where \(\mu\) represents the population mean for Ultrapower batteries.
You should assume that the population is Normally distributed with standard deviation 3.4.
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