5.05c Hypothesis test: normal distribution for population mean

7

1. The random variable \(X\) has a Poisson distribution with parameter \(\lambda\).
   1. Prove, from first principles, that \(\mathrm { E } ( X ) = \lambda\).
   2. Given that \(\mathrm { E } \left( X ^ { 2 } - X \right) = \lambda ^ { 2 }\), deduce that \(\operatorname { Var } ( X ) = \lambda\).
2. The number of faults in a 100-metre ball of nylon string may be modelled by a Poisson distribution with parameter \(\lambda\).
   1. An analysis of one ball of string, selected at random, showed 15 faults. Using an exact test, investigate the claim that \(\lambda > 10\). Use the \(5 \%\) level of significance.
   2. A subsequent analysis of a random sample of 20 balls of string showed a total of 241 faults.
      (A) Using an approximate test, re-investigate the claim that \(\lambda > 10\). Use the \(5 \%\) level of significance.
      (B) Determine the critical value of the total number of faults for the test in part (b)(ii)(A).
      (C) Given that, in fact, \(\lambda = 12\), estimate the probability of a Type II error for a test of the claim that \(\lambda > 10\) based upon a random sample of 20 balls of string and using the \(5 \%\) level of significance.
      [0pt] [4 marks] \includegraphics[max width=\textwidth, alt={}, center]{d5852425-9340-4aae-82da-e3bf6772a0de-22_2490_1728_219_141} \includegraphics[max width=\textwidth, alt={}, center]{d5852425-9340-4aae-82da-e3bf6772a0de-23_2490_1719_217_150} \includegraphics[max width=\textwidth, alt={}, center]{d5852425-9340-4aae-82da-e3bf6772a0de-24_2489_1728_221_141}

4

1. A large survey in the USA establishes that 60 per cent of its residents own a smartphone. A survey of 250 UK residents reveals that 164 of them own a smartphone.
   Assuming that these 250 UK residents may be regarded as a random sample, investigate the claim that the percentage of UK residents owning a smartphone is the same as that in the USA. Use the 5\% level of significance.
2. A random sample of 40 residents in a market town reveals that 5 of them own a 4 G mobile phone. Use an exact test to investigate, at the \(5 \%\) level of significance, the belief that fewer than 25 per cent of the town's residents own a 4 G mobile phone.
3. A marketing company needs to estimate the proportion of residents in a large city who own a 4 G mobile phone. It wishes to estimate this proportion to within 0.05 with a confidence of 98\%. Given that the proportion is known to be at most 30 per cent, estimate the sample size necessary in order to meet the company's need.
   [0pt] [5 marks]

6. The weight of a particular electrical component is normally distributed with a mean of 46.7 grams and a variance of 1.8 grams \(^ { 2 }\). The component is sold in boxes of 12 .

1. State the distribution of the mean weight of the components in one box.
2. Find the probability that the mean weight of the components in a randomly chosen box is more than 47 grams.
   (3 marks)
   After a break in production the component manufacturer wishes to find out if the mean weight of the components has changed. A random sample of 30 components is found to have a mean weight of 46.5 grams.
3. Assuming that the variance of the weight of the components is unchanged, test at the \(5 \%\) level of significance if there has been any change in the mean weight of the components.
   (7 marks)

7. A telephone company believes that, for young people, the average length of a telephone call on a land line is longer than on a mobile, due to the difference in price. The company collected data on the time, \(t\) minutes, of 500 calls made by young people on mobiles and the data is summarised by $$\Sigma t = 7335 , \quad \Sigma t ^ { 2 } = 172040 .$$

1. Calculate unbiased estimates of the mean and variance of \(t\). For 200 calls made on land lines by the same young people, unbiased estimates of the mean and variance of the call length were 15.9 minutes and 108.5 minutes \({ } ^ { 2 }\) respectively.
2. Stating your hypotheses clearly, test at the \(5 \%\) level whether or not there is evidence that longer calls are made on land lines than on mobiles.
   (9 marks)
3. Explain the importance of the central limit theorem in carrying out the test in part (b).

1. The random variable \(X\) has an \(F\)-distribution with 8 and 12 degrees of freedom.

Find \(\mathrm { P } \left( \frac { 1 } { 5.67 } < X < 2.85 \right)\).
(4)

2. A mechanic is required to change car tyres. An inspector timed a random sample of 20 tyre changes and calculated the unbiased estimate of the population variance to be 6.25 minutes \({ } ^ { 2 }\). Test, at the \(5 \%\) significance level, whether or not the standard deviation of the population of times taken by the mechanic is greater than 2 minutes. State your hypotheses clearly.
(6)

6. A supervisor wishes to cheek the typing speed of a new typist. On 10 randomly selected occasions, the supervisor records the time taken for the new typist to type 100 words. The results, in seconds, are given below. $$110,125,130,126,128,127,118,120,122,125$$ The supervisor assumes that the time taken to type 100 words is normally distributed.

1. Calculate a 95\% confidence interval for
   1. the mean,
   2. the variance
      of the population of times taken by this typist to type 100 words. The supervisor requires the average time needed to type 100 words to be no more than 130 seconds and the standard deviation to be no more than 4 seconds.
2. Comment on whether or not the supervisor should be concerned about the speed of the new typist.

7. A grocer receives deliveries of cauliflowers from two different growers, \(A\) and \(B\). The grocer takes random samples of cauliflowers from those supplied by each grower. He measures the weight \(x\), in grams, of each cauliflower. The results are summarised in the table below.

1. Show, at the \(10 \%\) significance level, that the variances of the populations from which the samples are drawn can be assumed to be equal by testing the hypothesis \(\mathrm { H } _ { 0 } : \sigma _ { A } ^ { 2 } = \sigma _ { B } ^ { 2 }\) against hypothesis \(\mathrm { H } _ { 1 } : \sigma _ { A } ^ { 2 } \neq \sigma _ { B } ^ { 2 }\).
   (You may assume that the two samples come from normal populations.)
   (6) The grocer believes that the mean weight of cauliflowers provided by \(B\) is at least 150 g more than the mean weight of cauliflowers provided by \(A\).
2. Use a \(5 \%\) significance level to test the grocer's belief.
3. Justify your choice of test.

The random variable \(X\) has a \(\chi ^ { 2 }\)-distribution with 9 degrees of freedom.

1. Find \(\mathrm { P } ( 2.088 < X < 19.023 )\). The random variable \(Y\) follows an \(F\)-distribution with 12 and 5 degrees of freedom.
2. Find the upper and lower \(5 \%\) critical values for \(Y\).
   (3)
   (Total 6 marks)
   

2. The standard deviation of the length of a random sample of 8 fence posts produced by a timber yard was 8 mm . A second timber yard produced a random sample of 13 fence posts with a standard deviation of 14 mm .

1. Test, at the \(10 \%\) significance level, whether or not there is evidence that the lengths of fence posts produced by these timber yards differ in variability. State your hypotheses clearly.
2. State an assumption you have made in order to carry out the test in part (a).

3. A machine is set to fill bags with flour such that the mean weight is 1010 grams. To check that the machine is working properly, a random sample of 8 bags is selected. The weight of flour, in grams, in each bag is as follows. $$\begin{array} { l l l l l l l l } 1010 & 1015 & 1005 & 1000 & 998 & 1008 & 1012 & 1007 \end{array}$$ Carry out a suitable test, at the \(5 \%\) significance level, to test whether or not the mean weight of flour in the bags is less than 1010 grams. (You may assume that the weight of flour delivered by the machine is normally distributed.)
(Total 8 marks)

6. Brickland and Goodbrick are two manufacturers of bricks. The lengths of the bricks produced by each manufacturer can be assumed to be normally distributed. A random sample of 20 bricks is taken from Brickland and the length, \(x \mathrm {~mm}\), of each brick is recorded. The mean of this sample is 207.1 mm and the variance is \(3.2 \mathrm {~mm} ^ { 2 }\).

1. Calculate the \(98 \%\) confidence interval for the mean length of brick from Brickland. A random sample of 10 bricks is selected from those manufactured by Goodbrick. The length of each brick, \(y \mathrm {~mm}\), is recorded. The results are summarised as follows. $$\sum y = 2046.2 \quad \sum y ^ { 2 } = 418785.4$$ The variances of the length of brick for each manufacturer are assumed to be the same.
2. Find a \(90 \%\) confidence interval for the value by which the mean length of brick made by Brickland exceeds the mean length of brick made by Goodbrick.

1. Historical records from a large colony of squirrels show that the weight of squirrels is normally distributed with a mean of 1012 g . Following a change in the diet of squirrels, a biologist is interested in whether or not the mean weight has changed.

A random sample of 14 squirrels is weighed and their weights \(x\), in grams, recorded. The results are summarised as follows: $$\Sigma x = 13700 , \quad \Sigma x ^ { 2 } = 13448750 .$$ Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there has been a change in the mean weight of the squirrels.

2. The weights, in grams, of apples are assumed to follow a normal distribution. The weights of apples sold by a supermarket have variance \(\sigma _ { s } { } ^ { 2 }\). A random sample of 4 apples from the supermarket had weights $$\text { 114, 100, 119, } 123 .$$

1. Find a 95\% confidence interval for \(\sigma _ { s } ^ { 2 }\). The weights of apples sold on a market stall have variance \(\sigma _ { M } ^ { 2 }\). A second random sample of 7 apples was taken from the market stall. The sample variance \(s _ { M } ^ { 2 }\) of the apples was 318.8.
2. Stating your hypotheses clearly test, at the \(1 \%\) levcel of significnace, whether or not there is evidence that \(\sigma _ { M } ^ { 2 } > \sigma _ { s } ^ { 2 }\).

4. Two machines \(A\) and \(B\) produce the same type of component in a factory. The factory manager wishes to know whether the lengths, \(x \mathrm {~cm}\), of the components produced by the two machines have the same mean. The manager took a random sample of components from each machine and the results are summarised in the table below. The lengths of components produced by the machines can be assumed to follow normal distributions.

1. Use a two tail test to show, at the \(10 \%\) significance level, that the variances of the lengths of components produced by each machine can be assumed to be equal.
   (4)
2. Showing your working clearly, find a \(95 \%\) confidence interval for \(\mu _ { B } - \mu _ { A }\), where \(\mu _ { A }\) and \(\mu _ { B }\) are the mean lengths of the populations of components produced by machine \(A\) and machine \(B\) respectively. There are serious consequences for the production at the factory if the difference in mean lengths of the components produced by the two machines is more than 0.7 cm .
3. State, giving your reason, whether or not the factory manager should be concerned.

2. The value of orders, in \(\pounds\), made to a firm over the internet has distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). A random sample of \(n\) orders is taken and \(\bar { X }\) denotes the sample mean.

1. Write down the mean and variance of \(\bar { X }\) in terms of \(\mu\) and \(\sigma ^ { 2 }\). A second sample of \(m\) orders is taken and \(\bar { Y }\) denotes the mean of this sample.
   An estimator of the population mean is given by $$U = \frac { n \bar { X } + m \bar { Y } } { n + m }$$
2. Show that \(U\) is an unbiased estimator for \(\mu\).
3. Show that the variance of \(U\) is \(\frac { \sigma ^ { 2 } } { n + m }\).
4. State which of \(\bar { X }\) or \(U\) is a better estimator for \(\mu\). Give a reason for your answer.

3. The lengths, \(x \mathrm {~mm}\), of the forewings of a random sample of male and female adult butterflies are measured. The following statistics are obtained from the data.

1. Assuming the lengths of the forewings are normally distributed test, at the \(10 \%\) level of significance, whether or not the variances of the two distributions are the same. State your hypotheses clearly.
2. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether the mean length of the forewings of the female butterflies is less than the mean length of the forewings of the male butterflies.
   (6)

4. The length \(X \mathrm {~mm}\) of a spring made by a machine is normally distributed \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). A random sample of 20 springs is selected and their lengths measured in mm . Using this sample the unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\) are $$\bar { x } = 100.6 , \quad s ^ { 2 } = 1.5 .$$ Stating your hypotheses clearly test, at the \(10 \%\) level of significance,

1. whether or not the variance of the lengths of springs is different from 0.9 ,
2. whether or not the mean length of the springs is greater than 100 mm .

5. The number of tornadoes per year to hit a particular town follows a Poisson distribution with mean \(\lambda\). A weatherman claims that due to climate changes the mean number of tornadoes per year has decreased. He records the number of tornadoes \(x\) to hit the town last year. To test the hypotheses \(\mathrm { H } _ { 0 } : \lambda = 7\) and \(\mathrm { H } _ { 1 } : \lambda < 7\), a critical region of \(x \leq 3\) is used.

1. Find, in terms \(\lambda\) the power function of this test.
2. Find the size of this test.
3. Find the probability of a Type II error when \(\lambda = 4\).

6. A butter packing machine cuts butter into blocks. The weight of a block of butter is normally distributed with a mean weight of 250 g and a standard deviation of 4 g . A random sample of 15 blocks is taken to monitor any change in the mean weight of the blocks of butter.

1. Find the critical region of a suitable test using a \(2 \%\) level of significance.
   (3)
2. Assuming the mean weight of a block of butter has increased to 254 g , find the probability of a Type II error.

7. A doctor wishes to study the level of blood glucose in males. The level of blood glucose is normally distributed. The doctor measured the blood glucose of 10 randomly selected male students from a school. The results, in mmol/litre, are given below. $$\begin{array} { l l l l l l l l l l } 4.7 & 3.6 & 3.8 & 4.7 & 4.1 & 2.2 & 3.6 & 4.0 & 4.4 & 5.0 \end{array}$$

1. Calculate a \(95 \%\) confidence interval for the mean.
2. Calculate a 95\% confidence interval for the variance. A blood glucose reading of more than 7 mmol/litre is counted as high.
3. Use appropriate confidence limits from parts (a) and (b) to find the highest estimate of the proportion of male students in the school with a high blood glucose level. \section*{END}

1. A large number of students are split into two groups \(A\) and \(B\). The students sit the same test but under different conditions. Group A has music playing in the room during the test, and group B has no music playing during the test. Small samples are then taken from each group and their marks recorded. The marks are normally distributed.

The marks are as follows:

1. Stating your hypotheses clearly, and using a \(10 \%\) level of significance, test whether or not there is evidence of a difference between the variances of the marks of the two groups.
2. State clearly an assumption you have made to enable you to carry out the test in part (a).
3. Use a two tailed test, with a \(5 \%\) level of significance, to determine if the playing of music during the test has made any difference in the mean marks of the two groups. State your hypotheses clearly.
4. Write down what you can conclude about the effect of music on a student's performance during the test.

1. The weights, in grams, of mice are normally distributed. A biologist takes a random sample of 10 mice. She weighs each mouse and records its weight.

The ten mice are then fed on a special diet. They are weighed again after two weeks.
Their weights in grams are as follows: Stating your hypotheses clearly, and using a \(1 \%\) level of significance, test whether or not the diet causes an increase in the mean weight of the mice.

A drug is claimed to produce a cure to a certain disease in \(35 \%\) of people who have the disease. To test this claim a sample of 20 people having this disease is chosen at random and given the drug. If the number of people cured is between 4 and 10 inclusive the claim will be accepted. Otherwise the claim will not be accepted.

1. Write down suitable hypotheses to carry out this test.
2. Find the probability of making a Type I error. The table below gives the value of the probability of the Type II error, to 4 decimal places, for different values of \(p\) where \(p\) is the probability of the drug curing a person with the disease.

   P (cure)	0.2	0.3	0.4	0.5
   P (Type II error)	0.5880	\(r\)	0.8565	\(s\)

3. Calculate the value of \(r\) and the value of \(s\).
4. Calculate the power of the test for \(p = 0.2\) and \(p = 0.4\)
5. Comment, giving your reasons, on the suitability of this test procedure.

1. An engineering firm buys steel rods. The steel rods from its present supplier are known to have a mean tensile strength of \(230 \mathrm {~N} / \mathrm { mm } ^ { 2 }\).

A new supplier of steel rods offers to supply rods at a cheaper price than the present supplier. A random sample of ten rods from this new supplier gave tensile strengths, \(x \mathrm { N } / \mathrm { mm } ^ { 2 }\), which are summarised below.

1. Stating your hypotheses clearly, and using a \(5 \%\) level of significance, test whether or not the rods from the new supplier have a tensile strength lower that the present supplier. (You may assume that the tensile strength is normally distributed).
2. In the light of your conclusion to part (a) write down what you would recommend the engineering firm to do.