Known variance (z-distribution)

1 The weights, in grams, of packets of sugar are distributed with mean \(\mu\) and standard deviation 23. A random sample of 150 packets is taken. The mean weight of this sample is found to be 494 g . Calculate a 98\% confidence interval for \(\mu\).

1 The heights of a certain species of deer are known to have standard deviation 0.35 m . A zoologist takes a random sample of 150 of these deer and finds that the mean height of the deer in the sample is 1.42 m .

1. Calculate a 96\% confidence interval for the population mean height.
2. Bubay says that \(96 \%\) of deer of this species are likely to have heights that are within this confidence interval. Explain briefly whether Bubay is correct.

1 Packets of fish food have weights that are distributed with standard deviation 2.3 g . A random sample of 200 packets is taken. The mean weight of this sample is found to be 99.2 g . Calculate a \(99 \%\) confidence interval for the population mean weight.

1 A random variable has a normal distribution with unknown mean \(\mu\) and known standard deviation 0.19 . In order to estimate \(\mu\) a random sample of five observations of the random variable was taken. The values were as follows. $$\begin{array} { l l l l l } 5.44 & 4.93 & 5.12 & 5.36 & 5.40 \end{array}$$ Using these five values, calculate,

1. an estimate of \(\mu\),
2. a 95\% confidence interval for \(\mu\).

11 Zac is planning to write a report on the music preferences of the students at his college. There is a large number of students at the college.

1. State one reason why Zac might wish to obtain information from a sample of students, rather than from all the students.
2. Amaya suggests that Zac should use a sample that is stratified by school year. Give one advantage of this method as compared with random sampling, in this context. Zac decides to take a random sample of 60 students from his college. He asks each student how many hours per week, on average, they spend listening to music during term. From his results he calculates the following statistics.

   Mean	Standard deviation	Median	Lower quartile	Upper quartile
   21.0	4.20	20.5	18.0	22.9

3. Sundip tells Zac that, during term, she spends on average 30 hours per week listening to music. Discuss briefly whether this value should be considered an outlier.
4. Layla claims that, during term, each student spends on average 20 hours per week listening to music. Zac believes that the true figure is higher than 20 hours. He uses his results to carry out a hypothesis test at the 5\% significance level. Assume that the time spent listening to music is normally distributed with standard deviation 4.20 hours. Carry out the test.

3. Athletes who compete in the 400 m event have resting heart rates (RHR), measured in beats per minute, which are normally distributed with known standard deviation \(4 \cdot 7\). A random sample of 90 athletes who compete in the 400 m event is taken. Their resting heart rates are summarised by $$\sum x = 4014 \quad \text { and } \quad \sum x ^ { 2 } = 182257 .$$

1. Find a \(99 \%\) confidence interval for the mean of the RHR of athletes who compete in the 400 m event. Give the limits of your interval correct to 1 decimal place.
2. Without doing any further calculation, explain how the width of a \(95 \%\) confidence interval would compare to the width of your interval in part (a). Athletes who compete in the discus event have RHR which are normally distributed with known standard deviation \(\sigma\). A random sample of 100 athletes who compete in the discus event is taken. A 95\% confidence interval for the mean of the RHR is calculated as [49•4, 52•6].
3. Determine the value of \(\sigma\) that was used to calculate this confidence interval.
4. Referring to the confidence intervals, state, with a reason, what can be said about the RHR of athletes who compete in the 400 m event compared to the RHR of athletes who compete in the discus event.

1. A factory produces yellow tennis balls and white tennis balls. Independent samples, one of yellow tennis balls and one of white tennis balls, are taken. The table shows information about the weights of the yellow tennis balls, \(Y\) grams, and the weights of the white tennis balls, \(W\) grams.

1. Find a 95\% confidence interval for the mean weight of yellow tennis balls. Jamie claims that the mean weight of the population of yellow tennis balls is greater than the mean weight of the population of white tennis balls. A test of Jamie's claim is carried out.
2. 1. Specify the approximate distribution of \(\bar { Y } - \bar { W }\) under the null hypothesis of the test.
   2. Explain the relevance of the large sample sizes to your answer to part (i).
3. Complete the hypothesis test using a \(5 \%\) level of significance. You should state your hypotheses and the value of your test statistic clearly.

1. Camilo grows two types of apple, green apples and red apples.

The standard deviation of the weights of green apples is known to be 3.5 grams.
A random sample of 80 green apples has a mean weight of 128 grams.

1. Find a 98\% confidence interval for the mean weight of the population of green apples. Show your working clearly and give the confidence interval limits to 2 decimal places. Camilo believes that the mean weight of the population of green apples is more than 10 grams greater than the mean weight of the population of red apples. A random sample of \(n\) red apples has a mean weight of 117 grams.
   The standard deviation of the weights of the red apples is known to be 4 grams.
   A test of Camilo's belief is carried out at the 5\% level of significance.
2. State the null and alternative hypotheses for this test.
3. Find the smallest value of \(n\) for which the null hypothesis will be rejected.
4. Explain the relevance of the Central Limit Theorem in parts (a) and (c).
5. Given that \(n = 85\), state the conclusion of the hypothesis test.

1. A factory produces bolts. The lengths of the bolts are normally distributed with mean \(\mu \mathrm { mm }\) and standard deviation 0.868 mm

A random sample of 15 of these bolts is taken and the mean length is 30.03 mm

1. Calculate a 90\% confidence interval for \(\mu\) A suitable test, at the \(10 \%\) level of significance, is carried out using these 15 bolts, to see whether or not there is evidence that the variance of the length of the bolts has increased.
2. Calculate the critical region for \(S ^ { 2 }\) The manager of the factory decides that, in future, he will check each month whether the machine making the bolts is working properly. He uses a \(10 \%\) level of significance to test whether or not there is evidence that
   • the mean length of the bolts has changed
   • the variance of the length of the bolts has increased
   The next month a random sample of 15 bolts is taken.
   The mean length of these bolts is 30.06 mm and the standard deviation is 1.02 mm
3. With reference to your answers to part (a) and part (b), state whether or not there is any evidence that the machine is not working properly.
   Give reasons for your answer.

4 The waiting times for patients to see a doctor in a hospital can be modelled with a normal distribution with known variance of 10 minutes. 4

1. A random sample of 100 patients has a total waiting time of 3540 minutes.
   Calculate a \(98 \%\) confidence interval for the population mean of waiting times, giving values to four significant figures.
   4
2. Dante conducts a hypothesis test with the sample from part (a) on the waiting times. Dante's hypotheses are $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 38 \\ & \mathrm { H } _ { 1 } : \mu \neq 38 \end{aligned}$$ Dante uses a \(2 \%\) level of significance.
   Explain whether Dante accepts or rejects the null hypothesis.

1
The continuous random variable \(X\) has the distribution \(\mathrm { N } ( \mu , 30 )\). The mean of a random sample of 8 observations of \(X\) is 53.1 . Determine a \(95 \%\) confidence interval for \(\mu\). You should give the end points of the interval correct to 4 significant figures.