Z-tests (known variance)

7 The mean weight of bags of carrots is \(\mu\) kilograms. An inspector wishes to test whether \(\mu = 2.0\). He weighs a random sample of 200 bags and his results are summarised as follows. $$\Sigma x = 430 \quad \Sigma x ^ { 2 } = 1290$$ Carry out the test at the 10\% significance level.

2 Karim has noted the lifespans, in weeks, of a large random sample of certain insects. He carries out a test, at the \(1 \%\) significance level, for the population mean, \(\mu\). Karim's null hypothesis is \(\mu = 6.4\).

1. Given that Karim's test is two-tail, state the alternative hypothesis.
   Karim finds that the value of the test statistic is \(z = 2.43\).
2. Explain what conclusion he should draw.
3. Explain briefly when a one-tail test is appropriate, rather than a two-tail test.

13 Each weekday Keira drives to work with her son Kaito. She always sets off at 8.00 a.m. She models her journey time, \(x\) minutes, by the distribution \(X \sim \mathrm {~N} ( 15,4 )\). Over a long period of time she notes that her journey takes less than 14 minutes on \(7 \%\) of the journeys, and takes more than 18 minutes on \(31 \%\) of the journeys.

1. Investigate whether Keira's model is a good fit for the data. Kaito believes that Keira's value for the variance is correct, but realises that the mean is not correct.
2. Find, correct to two significant figures, the value of the mean that Keira should use in a refined model which does fit the data. Keira buys a new car. After driving to work in it each day for several weeks, she randomly selects the journey times for \(n\) of these days. Her mean journey time for these \(n\) days is 16 minutes. Using the refined model she conducts a hypothesis test to see if her mean journey time has changed, and finds that the result is significant at the \(5 \%\) level.
3. Determine the smallest possible value of \(n\).

5 The mean solubility rating of widgets inserted into beer cans is thought to be 84.0, in appropriate units. A random sample of 50 widgets is taken. The solubility ratings, \(x\), are summarised by $$n = 50 , \quad \Sigma x = 4070 , \quad \Sigma x ^ { 2 } = 336100$$ Test, at the \(5 \%\) significance level, whether the mean solubility rating is less than 84.0 .

2 Past experience has shown that the heights of a certain variety of plant have mean 64.0 cm and standard deviation 3.8 cm . During a particularly hot summer, it was expected that the heights of plants of this variety would be less than usual. In order to test whether this was the case, a botanist recorded the heights of a random sample of 100 plants and found that the value of the sample mean was 63.3 cm . Stating a necessary assumption, carry out the test at the \(2.5 \%\) significance level.

3 Batteries of type \(A\) are known to have a mean life of 150 hours. It is required to test whether a new type of battery, type \(B\), has a shorter mean life than type \(A\) batteries.

1. Give a reason for using a sample rather than the whole population in carrying out this test.
   A random sample of 120 type \(B\) batteries are tested and it is found that their mean life is 147 hours, and an unbiased estimate of the population variance is 225 hours \(^ { 2 }\).
2. Test, at the \(2 \%\) significance level, whether type \(B\) batteries have a shorter mean life than type \(A\) batteries.
3. Calculate a \(94 \%\) confidence interval for the population mean life of type \(B\) batteries.

1 Roger claims that, on average, his journey time from home to work each day is greater than 45 minutes. The times, \(x\) minutes, of 30 randomly selected journeys result in \(\bar { x } = 45.8\) and \(s ^ { 2 } = 4.8\).
Investigate Roger's claim at the \(1 \%\) level of significance.

3 An architect wishes to investigate whether the buildings in a certain city are higher, on average, than buildings in other cities. He takes a large random sample of buildings from the city and finds the mean height of the buildings in the sample. He calculates the value of the test statistic, \(z\), and finds that \(z = 2.41\).

1. Explain briefly whether he should use a one-tail test or a two-tail test.
2. Carry out the test at the \(1 \%\) significance level.

3 In the past, the annual amount of wheat produced per farm by a large number of similar sized farms in a certain region had mean 24.0 tonnes and standard deviation 5.2 tonnes. Last summer a new fertiliser was used by all the farms, and it was expected that the mean amount of wheat produced per farm would be greater than 24.0 tonnes. In order to test whether this was true, a scientist recorded the amounts of wheat produced by a random sample of 50 farms last summer. He found that the value of the sample mean was 25.8 tonnes. Stating a necessary assumption, carry out the test at the \(1 \%\) significance level.

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 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 \(\mu\).

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.

A random sample of 100 classical CDs produced by a record company had a mean playing time of 70.6 minutes and a standard deviation of 9.1 minutes. An independent random sample of 80 CDs produced by a different company had a mean playing time of 67.2 minutes with a standard deviation of 8.4 minutes.

1. Using a 1\% level of significance, test whether or not there is a difference in the mean playing times of the CDs produced by these two companies. State your hypotheses clearly. [8]
2. State an assumption you made in carrying out the test in part (a). [1]

5 Each of a random sample of 200 steel bars taken from a production line was examined and 27 were found to be faulty.

1. Find an approximate \(90 \%\) confidence interval for the proportion of faulty bars produced. A change in the production method was introduced which, it was claimed, would reduce the proportion of faulty bars. After the change, each of a further random sample of 100 bars was examined and 8 were found to be faulty.
2. Test the claim, at the \(10 \%\) significance level.

Records show that the distance driven by a bus driver in a week is normally distributed with mean 1150 km and standard deviation 105 km. New driving regulations are introduced and in the next 20 weeks he drives a total of 21 800 km.

1. Stating any assumption(s), test, at the 1% significance level, whether his mean weekly driving distance has decreased. [6]
2. A similar test at the 1% significance level was carried out using the data from another 20 weeks. State the probability of a Type I error and describe what is meant by a Type I error in this context. [2]

5 Members of a residents' association are concerned about the speeds of cars travelling through their village. They decide to record the speed, in mph , of each of a random sample of 10 cars travelling through their village, with the following results: $$\begin{array} { l l l l l l l l l l } 33 & 27 & 34 & 30 & 48 & 35 & 34 & 33 & 43 & 39 \end{array}$$

1. Construct a \(99 \%\) confidence interval for \(\mu\), the mean speed of cars travelling through the village, stating any assumption that you make.
2. Comment on the claim that a 30 mph speed limit is being adhered to by most motorists.
   (3 marks)

In the past, the time for Jeff's journey to work had mean 45.7 minutes and standard deviation 5.6 minutes. This year he is trying a new route. In order to test whether the new route has reduced his journey time, Jeff finds the mean time for a random sample of 30 journeys using the new route. He carries out a hypothesis test at the 2.5% significance level. Jeff assumes that, for the new route, the journey time has a normal distribution with standard deviation 5.6 minutes.

1. State appropriate null and alternative hypotheses for the test. [2]
2. Determine the rejection region for the test. [4]

6 Random samples of 200 'Alpha' and 150 'Beta' vacuum cleaners were monitored for reliability. It was found that 62 Alpha and 35 Beta cleaners required repair during the guarantee period of one year. The proportions of all Alpha and Beta cleaners that require repair during the guarantee period are \(p _ { \alpha }\) and \(p _ { \beta }\) respectively.

1. Find a \(95 \%\) confidence interval for \(p _ { \alpha }\).
2. Give a reason why, apart from rounding, the interval is approximate.
3. Test, at the \(5 \%\) significance level, whether \(p _ { \alpha }\) differs from \(p _ { \beta }\).

A doctor wishes to investigate the mean fat content in low-fat burgers. He takes a random sample of 15 burgers and sends them to a laboratory where the mass, in grams, of fat in each burger is determined. The results are as follows. \(9 \quad 7 \quad 8 \quad 9 \quad 6 \quad 11 \quad 7 \quad 9 \quad 8 \quad 9 \quad 8 \quad 10 \quad 7 \quad 9 \quad 9\) Assume that the mass, in grams, of fat in low-fat burgers is normally distributed with mean \(\mu\) and that the population standard deviation is 1.3.

1. Calculate a 99\% confidence interval for \(\mu\). [4]
2. Explain whether it was necessary to use the Central Limit theorem in the calculation in part (i). [2]
3. The manufacturer claims that the mean mass of fat in burgers of this type is 8 g. Use your answer to part (i) to comment on this claim. [2]

3 Ten randomly chosen athletes were coached for a 200 m event. For each athlete, the times taken to run 200 m before and after coaching were measured. The sample mean times before and after coaching were 23.43 seconds and 22.84 seconds respectively. For each athlete the difference, \(d\) seconds, in the times before and after coaching was calculated and an unbiased estimate of the population variance of \(d\) was found to be 0.548 . Stating any required assumption, test at the \(5 \%\) significance level whether the population mean time for the 200 m run decreased after coaching.

1 The result of a memory test is known to be normally distributed with mean \(\mu\) and standard deviation 1.9. It is required to have a \(95 \%\) confidence interval for \(\mu\) with a total width of less than 2.0 . Find the least possible number of tests needed to achieve this.

2 A six-sided die has faces marked \(1,2,3,4,5,6\). When the die is thrown 300 times it shows a six on 56 throws.

1. Calculate an approximate \(96 \%\) confidence interval for the probability that the die shows a six on one throw.
2. Maroulla claims that the die is biased. Use your answer to part (a) to comment on this claim.

3 A random sample of 500 households in a certain town was chosen. Using this sample, a confidence interval for the proportion, \(p\), of all households in that town that owned two or more cars was found to be \(0.355 < p < 0.445\). Find the confidence level of this confidence interval. Give your answer correct to the nearest integer.

1 The diameters, \(x\) millimetres, of a random sample of 200 discs made by a certain machine were recorded. The results are summarised below. $$n = 200 \quad \Sigma x = 2520 \quad \Sigma x ^ { 2 } = 31852$$

1. Calculate a 95\% confidence interval for the population mean diameter.
2. Jean chose 40 random samples and used each sample to calculate a 95\% confidence interval for the population mean diameter. How many of these 40 confidence intervals would be expected to include the true value of the population mean diameter?

1 A basketball club has a large number of players. The heights, \(x \mathrm {~m}\), of a random sample of 10 of these players are measured. A \(90 \%\) confidence interval for the population mean height, \(\mu \mathrm { m }\), of players in this club is calculated. It is assumed that heights are normally distributed. The confidence interval is \(1.78 \leqslant \mu \leqslant 2.02\). Find the values of \(\sum x\) and \(\sum x ^ { 2 }\) for this sample.

6 The independent random variables \(X\) and \(Y\) have distributions with the same variance \(\sigma ^ { 2 }\). Random samples of 5 observations of \(X\) and \(n\) observations of \(Y\) are made and the results are summarised by $$\Sigma x = 5.5 , \quad \Sigma x ^ { 2 } = 15.05 , \quad \Sigma y = 8.0 , \quad \Sigma y ^ { 2 } = 36.4$$ Given that the pooled estimate of \(\sigma ^ { 2 }\) is 3 , find the value of \(n\).

2 In the past the yield of a certain crop, in tonnes per hectare, had mean 0.56 and standard deviation 0.08 Following the introduction of a new fertilizer, the farmer intends to test at the \(2.5 \%\) significance level whether the mean yield has increased. He finds that the mean yield over 10 years is 0.61 tonnes per hectare.

1. State two assumptions that are necessary for the test.
2. Carry out the test.

1 Roger claims that, on average, his journey time from home to work each day is greater than 45 minutes. The times, \(x\) minutes, of 30 randomly selected journeys result in \(\bar { x } = 45.8\) and \(s ^ { 2 } = 4.8\).
Investigate Roger's claim at the \(1 \%\) level of significance.

3 A new treatment of cotton thread, designed to increase the breaking strength, was tested on a random sample of 6 pieces of a standard length. The breaking strengths, in grams, were as follows. $$\begin{array} { l l l l l l } 17.3 & 18.4 & 18.6 & 17.2 & 17.5 & 19.3 \end{array}$$ The breaking strengths of a random sample of 5 similar pieces of the thread which had not been treated were as follows. \section*{\(\begin{array} { l l l l l } 18.6 & 17.2 & 16.3 & 17.4 & 16.8 \end{array}\)} A test of whether the treatment has been successful is to be carried out.

1. State what distributional assumptions are needed.
2. Carry out the test at the \(10 \%\) significance level.

1 The lengths, \(X\) centimetres, of a random sample of 7 leaves from a certain variety of tree are as follows.
3.9
4.8
4.8
4.4

1. Kaff coffee is sold in packets. A seller measures the masses of the contents of a random sample of 90 packets of Kaff coffee from her stock. The results are shown in the table below.

$$\text { (You may use } \sum \mathrm { fy } ^ { 2 } = 5644 \text { 171.75) }$$ A histogram is drawn and the class \(245 \leq w < 248\) is represented by a rectangle of width 1.2 cm and height 10 cm .

1. Calculate the width and the height of the rectangle representing the class \(255 \leq w < 260\).
2. Use linear interpolation to estimate the median mass of the contents of a packet of Kaff coffee to 1 decimal place.
3. Estimate the mean and the standard deviation of the mass of the contents of a packet of Kaff coffee to 1 decimal place. The seller claims that the mean mass of the contents of the packets is more than the stated mass. Given that the stated mass of the contents of a packet of Kaff coffee is 250 g and the actual standard deviation of the contents of a packet of Kaff coffee is 4 g ,
4. test, using a 5\% level of significance, whether or not the seller's claim is justified. State your hypotheses clearly.
   (You may assume that the mass of the contents of a packet is normally distributed.)
5. Using your answers to parts (b) and (c), comment on the assumption that the mass of the contents of a packet is normally distributed.
   (Total 14 marks)

2 A die is biased. The mean and variance of a random sample of 70 scores on this die are found to be 3.61 and 2.70 respectively. Calculate a \(95 \%\) confidence interval for the population mean score.

1 The result of a fitness trial is a random variable \(X\) which is normally distributed with mean \(\mu\) and standard deviation 2.4. A researcher uses the results from a random sample of 90 trials to calculate a \(98 \%\) confidence interval for \(\mu\). What is the width of this interval?