5.05c Hypothesis test: normal distribution for population mean

1. A company manufactures bolts with a mean diameter of 5 mm . The company wishes to check that the diameter of the bolts has not decreased. A random sample of 10 bolts is taken and the diameters, \(x \mathrm {~mm}\), of the bolts are measured. The results are summarised below.

$$\sum x = 49.1 \quad \sum x ^ { 2 } = 241.2$$ Using a \(1 \%\) level of significance, test whether or not the mean diameter of the bolts is less than 5 mm .
(You may assume that the diameter of the bolts follows a normal distribution.)

2. An emission-control device is tested to see if it reduces \(\mathrm { CO } _ { 2 }\) emissions from cars. The emissions from 6 randomly selected cars are measured with and without the device. The results are as follows.

1. State an assumption that needs to be made in order to carry out a \(t\)-test in this case.
2. State why a paired \(t\)-test is suitable for use with these data.
3. Using a \(5 \%\) level of significance, test whether or not there is evidence that the device reduces \(\mathrm { CO } _ { 2 }\) emissions from cars.
4. Explain, in context, what a type II error would be in this case.

Define, in terms of \(\mathrm { H } _ { 0 }\) and/or \(\mathrm { H } _ { 1 }\),

1. the size of a hypothesis test,
2. the power of a hypothesis test. The probability of getting a head when a coin is tossed is denoted by \(p\).
   This coin is tossed 12 times in order to test the hypotheses \(\mathrm { H } _ { 0 } : p = 0.5\) against \(\mathrm { H } _ { 1 } : p \neq 0.5\), using a 5\% level of significance.
3. Find the largest critical region for this test, such that the probability in each tail is less than 2.5\%.
4. Given that \(p = 0.4\)
   1. find the probability of a type II error when using this test,
   2. find the power of this test.
5. Suggest two ways in which the power of the test can be increased.
   

1. A farmer set up a trial to assess whether adding water to dry feed increases the milk yield of his cows. He randomly selected 22 cows. Thirteen of the cows were given dry feed and the other 9 cows were given the feed with water added. The milk yields, in litres per day, were recorded with the following results.

You may assume that the milk yield from cows given the dry feed and the milk yield from cows given the feed with water added are from independent normal distributions.

1. Test, at the \(10 \%\) level of significance, whether or not the variances of the populations from which the samples are drawn are the same. State your hypotheses clearly.
2. Calculate a \(95 \%\) confidence interval for the difference between the two mean milk yields.
3. Explain the importance of the test in part (a) to the calculation in part (b).
   

1. A teacher wishes to test whether playing background music enables students to complete a task more quickly. The same task was completed by 15 students, divided at random into two groups. The first group had background music playing during the task and the second group had no background music playing.
   The times taken, in minutes, to complete the task are summarised below.

You may assume that the times taken to complete the task by the students are two independent random samples from normal distributions.

1. Stating your hypotheses clearly, test, at the \(10 \%\) level of significance, whether or not the variances of the times taken to complete the task with and without background music are equal.
2. Find a 99\% confidence interval for the difference in the mean times taken to complete the task with and without background music. Experiments like this are often performed using the same people in each group.
3. Explain why this would not be appropriate in this case.
   

1. As part of an investigation, a random sample of 10 people had their heart rate, in beats per minute, measured whilst standing up and whilst lying down. The results are summarised below.

1. State one assumption that needs to be made in order to carry out a paired \(t\)-test.
2. Test, at the \(5 \%\) level of significance, whether or not there is any evidence that standing up increases people's mean heart rate by more than 5 beats per minute. State your hypotheses clearly.

A manager in a sweet factory believes that the machines are working incorrectly and the proportion \(p\) of underweight bags of sweets is more than \(5 \%\). He decides to test this by randomly selecting a sample of 5 bags and recording the number \(X\) that are underweight. The manager sets up the hypotheses \(\mathrm { H } _ { 0 } : p = 0.05\) and \(\mathrm { H } _ { 1 } : p > 0.05\) and rejects the null hypothesis if \(x > 1\).

1. Find the size of the test.
2. Show that the power function of the test is $$1 - ( 1 - p ) ^ { 4 } ( 1 + 4 p )$$ The manager goes on holiday and his deputy checks the production by randomly selecting a sample of 10 bags of sweets. He rejects the hypothesis that \(p = 0.05\) if more than 2 underweight bags are found in the sample.
3. Find the probability of a Type I error using the deputy's test. \section*{Question 3 continues on page 12} The table below gives some values, to 2 decimal places, of the power function for the deputy's test.

   \(p\)	0.10	0.15	0.20	0.25
   Power	0.07	\(s\)	0.32	0.47

4. Find the value of \(s\). The graph of the power function for the manager's test is shown in Figure 1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0bc6c296-9cbe-498b-89d9-c034b1b246e4-08_1157_1436_847_260} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure}
5. On the same axes, draw the graph of the power function for the deputy's test.
6. 1. State the value of \(p\) where these graphs intersect.
   2. Compare the effectiveness of the two tests if \(p\) is greater than this value. The deputy suggests that they should use his sampling method rather than the manager's.
7. Give a reason why the manager might not agree to this change.

1. A car manufacturer claims that, on a motorway, the mean number of miles per gallon for the Panther car is more than 70 . To test this claim a car magazine measures the number of miles per gallon, \(x\), of each of a random sample of 20 Panther cars and obtained the following statistics.

$$\bar { x } = 71.2 \quad s = 3.4$$ The number of miles per gallon may be assumed to be normally distributed.

1. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test the manufacturer's claim. The standard deviation of the number of miles per gallon for the Tiger car is 4 .
2. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not there is evidence that the variance of the number of miles per gallon for the Panther car is different from that of the Tiger car.
   

1. Two independent random samples \(X _ { 1 } , X _ { 2 } , \ldots , X _ { 7 }\) and \(Y _ { 1 } , Y _ { 2 } , Y _ { 3 } , Y _ { 4 }\) were taken from different normal populations with a common standard deviation \(\sigma\). The following sample statistics were calculated.

$$s _ { x } = 14.67 \quad s _ { y } = 12.07$$ Find the \(99 \%\) confidence interval for \(\sigma ^ { 2 }\) based on these two samples.

1. The weights of the contents of breakfast cereal boxes are normally distributed.

A manufacturer changes the style of the boxes but claims that the weight of the contents remains the same.
A random sample of 6 old style boxes had contents with the following weights (in grams). $$\begin{array} { l l l l l l } 512 & 503 & 514 & 506 & 509 & 515 \end{array}$$ The weights, \(y\) grams, of the contents of an independent random sample of 5 new style boxes gave $$\bar { y } = 504.8 \text { and } s _ { y } = 3.420$$

1. Use a two-tail test to show, at the \(10 \%\) level of significance, that the variances of the weights of the contents of the old and new style boxes can be assumed to be equal. State your hypotheses clearly.
2. Showing your working clearly, find a \(90 \%\) confidence interval for \(\mu _ { x } - \mu _ { y }\), where \(\mu _ { x }\) and \(\mu _ { y }\) are the mean weights of the contents of old and new style boxes respectively.
3. With reference to your confidence interval comment on the manufacturer's claim.
   

1. A machine produces components whose lengths are normally distributed with mean 102.3 mm and standard deviation 2.8 mm . After the machine had been serviced, a random sample of 20 components were tested to see if the mean and standard deviation had changed. The lengths, \(x \mathrm {~mm}\), of each of these 20 components are summarised as

$$\sum x = 2072 \quad \sum x ^ { 2 } = 214856$$

1. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not there is evidence of a change in standard deviation.
2. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not the mean length of the components has changed from the original value of 102.3 mm using
   1. a normal distribution,
   2. a \(t\) distribution.
3. Comment on the mean length of components produced after the service in the light of the tests from part (a) and part (b). Give a reason for your answer.
   

2. The time, \(t\) hours, that a typist can sit before incurring back pain is modelled by \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). A random sample of 30 typists gave unbiased estimates for \(\mu\) and \(\sigma ^ { 2 }\) as shown below. $$\hat { \mu } = 2.5 \quad s ^ { 2 } = 0.36$$

1. Find a 95\% confidence interval for \(\sigma ^ { 2 }\)
2. State with a reason whether or not the confidence interval supports the assertion that \(\sigma ^ { 2 } = 0.495\)

4. A company carries out an investigation into the strengths of rods from two different suppliers, Ardo and Bards. Independent random samples of rods were taken from each supplier and the force, \(x \mathrm { kN }\), needed to break each rod was recorded. The company wrote the results on a piece of paper but unfortunately spilt ink on it so some of the results can not be seen.
The paper with the results on is shown below. \includegraphics[max width=\textwidth, alt={}, center]{4f096806-33da-453f-a4c1-12be20d1a96d-08_435_1522_541_244}

1. 1. Use the data from Ardo to calculate an unbiased estimate, \(s _ { A } ^ { 2 }\), of the variance.
   2. Hence find an unbiased estimate, \(s _ { B } ^ { 2 }\), of the variance for the sample of 9 values from Bards.
2. Stating your hypotheses clearly, test at the \(10 \%\) level of significance whether or not there is a difference in variability of strength between the rods from Ardo and the rods from Bards.
   (You may assume the two samples come from independent normal distributions.)
3. Use a \(5 \%\) level of significance to test whether the mean strength of rods from Bards is more than 0.9 kN greater than the mean strength of rods from Ardo.
   (6)

1. A machine fills bottles with water. The amount of water in each bottle is normally distributed. To check the machine is working properly, a random sample of 12 bottles is selected and the amount of water, in ml, in each bottle is recorded. Unbiased estimates for the mean and variance are

$$\hat { \mu } = 502 \quad s ^ { 2 } = 5.6$$ Stating your hypotheses clearly, test at the 1\% level of significance

1. whether or not the mean amount of water in a bottle is more than 500 ml ,
2. whether or not the standard deviation of the amount of water in a bottle is less than 3 ml .
   

7. A machine produces bricks. The lengths, \(x \mathrm {~mm}\), of the bricks are distributed \(\mathrm { N } \left( \mu , 2 ^ { 2 } \right)\). At the start of each week a random sample of \(n\) bricks is taken to check the machine is working correctly.
A test is then carried out at the \(1 \%\) level of significance with $$\mathrm { H } _ { 0 } : \mu = 202 \text { and } \mathrm { H } _ { 1 } : \mu < 202$$

1. Find, in terms of \(n\), the critical region of the test. The probability of a type II error, when \(\mu = 200\), is less than 0.05
2. Find the minimum value of \(n\).

1. George owns a garage and he records the mileage of cars, \(x\) thousands of miles, between services. The results from a random sample of 10 cars are summarised below.

$$\sum x = 113.4 \quad \sum x ^ { 2 } = 1414.08$$ The mileage of cars between services is normally distributed and George believes that the standard deviation is 2.4 thousand miles. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not these data support George's belief.

2. Every 6 months some engineers are tested to see if their times, in minutes, to assemble a particular component have changed. The times taken to assemble the component are normally distributed. A random sample of 8 engineers was chosen and their times to assemble the component were recorded in January and in July. The data are given in the table below.

1. Calculate a \(95 \%\) confidence interval for the mean difference in times.
2. Use your confidence interval to state, giving a reason, whether or not there is evidence of a change in the mean time to assemble a component. State your hypotheses clearly.

3. An archaeologist is studying the compression strength of bricks at some ancient European sites. He took random samples from two sites \(A\) and \(B\) and recorded the compression strength of these bricks in appropriate units. The results are summarised below. It can be assumed that the compression strength of bricks is normally distributed.

1. Test, at the \(2 \%\) level of significance, whether or not there is evidence of a difference in the variances of compression strength of the bricks between these two sites. State your hypotheses clearly.
   (5) Site \(A\) is older than site \(B\) and the archaeologist claims that the mean compression strength of the bricks was greater at the younger site.
2. Stating your hypotheses clearly and using a \(1 \%\) level of significance, test the archaeologist's claim.
3. Explain briefly the importance of the test in part (a) to the test in part (b).

6. The carbon content, measured in suitable units, of steel is normally distributed. Two independent random samples of steel were taken from a refining plant at different times and their carbon content recorded. The results are given below. Sample A: \(\quad 1.5 \quad 0.9 \quad 1.3 \quad 1.2\) \(\begin{array} { l l l l l l l } \text { Sample } B : & 0.4 & 0.6 & 0.8 & 0.3 & 0.5 & 0.4 \end{array}\)

1. Stating your hypotheses clearly, carry out a suitable test, at the \(10 \%\) level of significance, to show that both samples can be assumed to have come from populations with a common variance \(\sigma ^ { 2 }\).
2. Showing your working clearly, find the \(99 \%\) confidence interval for \(\sigma ^ { 2 }\) based on both samples.

3. A farmer is investigating the milk yields of two breeds of cow. He takes a random sample of 9 cows of breed \(A\) and an independent random sample of 12 cows of breed \(B\). For a 5 day period he measures the amount of milk, \(x\) gallons, produced by each cow. The results are summarised in the table below. The amount of milk produced by each cow can be assumed to follow a normal distribution.

1. Use a two-tail test to show, at the \(10 \%\) level of significance, that the variances of the yields of the two breeds can be assumed to be equal. State your hypotheses clearly.
2. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not there is a difference in the mean yields of the two breeds of cow.
3. Explain briefly the importance of the test in part (a) for the test in part (b).

1. At the start of each academic year, a large college carries out a diagnostic test on a random sample of new students. Past experience has shown that the standard deviation of the scores on this test is 19.71

The admissions tutor claimed that the new students in 2013 would have more varied scores than usual. The scores for the students taking the test can be assumed to come from a normal distribution. A random sample of 10 new students was taken and the score \(x\), for each student was recorded. The data are summarised as \(\sum x = 619 \sum x ^ { 2 } = 42397\)

1. Stating your hypotheses clearly, and using a \(5 \%\) level of significance, test the admission tutor's claim. The admissions tutor decides that in future he will use the same hypotheses but take a larger sample of size 30 and use a significance level of 1\%.
2. Use the tables to show that, to 3 decimal places, the critical region for \(S ^ { 2 }\) is \(S ^ { 2 } > 664.281\)
3. Find the probability of a type II error using this test when the true value of the standard deviation is in fact 22.20
   

5. A large company has designed an aptitude test for new recruits. The score, \(S\), for an individual taking the test, has a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). In order to estimate \(\mu\) and \(\sigma\), a random sample of 15 new recruits were given the test and their scores, \(x\), are summarised as $$\sum x = 880 \quad \sum x ^ { 2 } = 54892$$

1. Calculate a 95\% confidence interval for
   1. \(\mu\),
   2. \(\sigma\). The company wants to ensure that no more than \(80 \%\) of new recruits pass the test.
2. Using values from your confidence intervals in part (a), estimate the lowest pass mark they should set.

1. A production line is designed to fill bottles with oil. The amount of oil placed in a bottle is normally distributed and the mean is set to 100 ml .

The amount of oil, \(x \mathrm { ml }\), in each of 8 randomly selected bottles is recorded, and the following statistics are obtained. $$\bar { x } = 92.875 \quad s = 8.3055$$ Malcolm believes that the mean amount of oil placed in a bottle is less than 100 ml .
Stating your hypotheses clearly, test, at the \(5 \%\) significance level, whether or not Malcolm's belief is supported.

3. A large number of chicks were fed a special diet for 10 days. A random sample of 9 of these chicks is taken and the weight gained, \(x\) grams, by each chick is recorded. The results are summarised below. $$\sum x = 181 \quad \sum x ^ { 2 } = 3913$$ You may assume that the weights gained by the chicks are normally distributed.
Calculate a 95\% confidence interval for

1. 1. the mean of the weights gained by the chicks,
   2. the variance of the weights gained by the chicks. A chick which gains less than \(16 g\) has to be given extra feed.
2. Using appropriate confidence limits from part (a), find the lowest estimate of the proportion of chicks that need extra feed.

7. Two groups of students take the same examination. A random sample of students is taken from each of the groups. The marks of the 9 students from Group 1 are as follows $$\begin{array} { l l l l l l l l l } 30 & 29 & 35 & 27 & 23 & 33 & 33 & 35 & 28 \end{array}$$ The marks, \(x\), of the 7 students from Group 2 gave the following statistics $$\bar { x } = 31.29 \quad s ^ { 2 } = 12.9$$ A test is to be carried out to see whether or not there is a difference between the mean marks of the two groups of students. You may assume that the samples are taken from normally distributed populations and that they are independent.

1. State one other assumption that must be made in order to apply this test and show that this assumption is reasonable by testing it at a \(10 \%\) level of significance. State your hypotheses clearly.
2. Stating your hypotheses clearly, test, using a significance level of \(5 \%\), whether or not there is a difference between the mean marks of the two groups of students.