5.05c Hypothesis test: normal distribution for population mean

A perfume manufacturer had one bottle-filling machine, but because of increased sales a second machine was obtained. In order to compare the performance of the two machines, a random sample of 50 bottles filled by the first machine and a random sample of 60 bottles filled by the second machine were checked. The volumes of the contents from the first machine, \(x _ { 1 } \mathrm { ml }\), and from the second machine, \(x _ { 2 } \mathrm { ml }\), are summarised by $$\Sigma x _ { 1 } = 1492.0 , \quad \Sigma x _ { 1 } ^ { 2 } = 44529.52 , \quad \Sigma x _ { 2 } = 1803.6 , \quad \Sigma x _ { 2 } ^ { 2 } = 54220.84 .$$ The volumes have distributions with means \(\mu _ { 1 } \mathrm { ml }\) and \(\mu _ { 2 } \mathrm { ml }\) for the first and second machines respectively. Test, at the \(2 \%\) significance level, whether \(\mu _ { 2 }\) is greater than \(\mu _ { 1 }\). Find the set of values of \(\alpha\) for which there would be evidence at the \(\alpha \%\) significance level that \(\mu _ { 2 } - \mu _ { 1 } > 0.1\).

9 The leaves from oak trees growing in two different areas \(A\) and \(B\) are being measured. The lengths, in cm , of a random sample of 7 oak leaves from area \(A\) are $$6.2 , \quad 8.3 , \quad 7.8 , \quad 9.3 , \quad 10.2 , \quad 8.4 , \quad 7.2$$ Assuming that the distribution is normal, find a 95\% confidence interval for the mean length of oak leaves from area \(A\). The lengths, in cm, of a random sample of 5 oak leaves from area \(B\) are $$5.9 , \quad 7.4 , \quad 6.8 , \quad 8.2 , \quad 8.7$$ Making suitable assumptions, which should be stated, test, at the \(5 \%\) significance level, whether the mean length of oak leaves from area \(A\) is greater than the mean length of oak leaves from area \(B\). [9]

9 Experiments are conducted to test the breaking strength of each of two types of rope, \(P\) and \(Q\). A random sample of 50 ropes of type \(P\) and a random sample of 70 ropes of type \(Q\) are selected. The breaking strengths, \(p\) and \(q\), measured in appropriate units, are summarised as follows. $$\Sigma p = 321.2 \quad \Sigma p ^ { 2 } = 2120.0 \quad \Sigma q = 475.3 \quad \Sigma q ^ { 2 } = 3310.0$$ Test, at the \(10 \%\) significance level, whether the mean breaking strengths of type \(P\) and type \(Q\) ropes are the same.

7 A random sample of 10 observations of a normally distributed random variable \(X\) gave the following summarised data, where \(\bar { x }\) denotes the sample mean. $$\Sigma x = 70.4 \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 8.48$$ Test, at the \(10 \%\) significance level, whether the population mean of \(X\) is less than 7.5.

A factory produces bottles of spring water. The manager decides to assess the performance of the two machines that are used to fill the bottles with water. He selects a random sample of 60 bottles filled by the first machine \(X\) and a random sample of 80 bottles filled by the second machine \(Y\). The volumes of water, \(x\) and \(y\), measured in appropriate units, are summarised as follows. $$\Sigma x = 58.2 \quad \Sigma x ^ { 2 } = 85.8 \quad \Sigma y = 97.6 \quad \Sigma y ^ { 2 } = 188.6$$ A test at the \(\alpha \%\) significance level shows that the mean volume of water in bottles filled by machine \(X\) is less than the mean volume of water in bottles filled by machine \(Y\). Find the set of possible values of \(\alpha\).

9 A random sample of 9 observations of a normally distributed random variable \(X\) gave the following summarised data. $$\Sigma x = 94.5 \quad \Sigma x ^ { 2 } = 993.6$$ Test, at the \(5 \%\) significance level, whether the population mean of \(X\) is 10.2 . Calculate a \(90 \%\) confidence interval for the population mean of \(X\).

6 A random sample of 50 observations of a random variable \(X\) and a random sample of 60 observations of a random variable \(Y\) are taken. The results for the sample means, \(\bar { x }\) and \(\bar { y }\), and the unbiased estimates for the population variances, \(s _ { x } ^ { 2 }\) and \(s _ { y } ^ { 2 }\), respectively, are as follows. $$\bar { x } = 25.4 \quad \bar { y } = 23.6 \quad s _ { x } ^ { 2 } = 23.2 \quad s _ { y } ^ { 2 } = 27.8$$ A test, at the \(\alpha \%\) significance level, of the null hypothesis that the population means of \(X\) and \(Y\) are equal against the alternative hypothesis that they are not equal is carried out. Given that the null hypothesis is not rejected, find the set of possible values of \(\alpha\).

A uniform plane object consists of three identical circular rings, \(X , Y\) and \(Z\), enclosed in a larger circular ring \(W\). Each of the inner rings has mass \(m\) and radius \(r\). The outer ring has mass \(3 m\) and radius \(R\). The centres of the inner rings lie at the vertices of an equilateral triangle of side \(2 r\). The outer ring touches each of the inner rings and the rings are rigidly joined together. The fixed axis \(A B\) is the diameter of \(W\) that passes through the centre of \(X\) and the point of contact of \(Y\) and \(Z\) (see diagram). It is given that \(R = \left( 1 + \frac { 2 } { 3 } \sqrt { } 3 \right) r\).

1. Show that the moment of inertia of the object about \(A B\) is \(( 7 + 2 \sqrt { } 3 ) m r ^ { 2 }\). The line \(C D\) is the diameter of \(W\) that is perpendicular to \(A B\). A particle of mass \(9 m\) is attached to \(D\). The object is now held with its plane horizontal. It is released from rest and rotates freely about the fixed horizontal axis \(A B\).
2. Find, in terms of \(g\) and \(r\), the angular speed of the object when it has rotated through \(60 ^ { \circ }\).

Fish of a certain species live in two separate lakes, \(A\) and \(B\). A zoologist claims that the mean length of fish in \(A\) is greater than the mean length of fish in \(B\). To test his claim, he catches a random sample of 8 fish from \(A\) and a random sample of 6 fish from \(B\). The lengths of the 8 fish from \(A\), in appropriate units, are as follows. $$\begin{array} { l l l l l l l l } 15.3 & 12.0 & 15.1 & 11.2 & 14.4 & 13.8 & 12.4 & 11.8 \end{array}$$ Assuming a normal distribution, find a \(95 \%\) confidence interval for the mean length of fish in \(A\). The lengths of the 6 fish from \(B\), in the same units, are as follows. $$\begin{array} { l l l l l l } 15.0 & 10.7 & 13.6 & 12.4 & 11.6 & 12.6 \end{array}$$ Stating any assumptions that you make, test at the \(5 \%\) significance level whether the mean length of fish in \(A\) is greater than the mean length of fish in \(B\). Calculate a 95\% confidence interval for the difference in the mean lengths of fish from \(A\) and from \(B\).

6 A random sample of 8 observations of a normal random variable \(X\) has mean \(\bar { x }\), where $$\bar { x } = 6.246 \quad \text { and } \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 0.784$$ Test, at the \(5 \%\) significance level, whether the population mean of \(X\) is less than 6.44.

8 The amounts spent on the weekly food shopping by families in the big city \(P\) and the small town \(Q\) are to be compared. The amounts spent, in dollars, in \(P\) and \(Q\) are denoted by \(x\) and \(y\) respectively. For a random sample of 60 families in \(P\) and a random sample of 50 families in \(Q\), the amounts are summarised as follows. $$\Sigma x = 9600 \quad \Sigma x ^ { 2 } = 1560000 \quad \Sigma y = 7200 \quad \Sigma y ^ { 2 } = 1052500$$ Assuming a common population variance, find

1. a pooled estimate for the population variance,
2. a \(95 \%\) confidence interval for the difference in the population means in \(P\) and \(Q\).

9 The land areas \(x\) (in suitable units) and populations \(y\) (in millions) for a sample of 8 randomly chosen cities are given in the following table. $$\left[ \Sigma x = 35.9 , \Sigma x ^ { 2 } = 216.47 , \Sigma y = 36.8 , \Sigma y ^ { 2 } = 244.96 , \Sigma x y = 212.62 . \right]$$

1. Find, showing all necessary working, the value of the product moment correlation coefficient for this sample.
2. Using a \(1 \%\) significance level, test whether there is positive correlation between land area and population of cities.
   The land areas and populations for another randomly chosen sample of cities, this time of size \(n\), give a product moment correlation coefficient of 0.651 . Using a test at the \(1 \%\) significance level, there is evidence of non-zero correlation between the variables.
3. Find the least possible value of \(n\), justifying your answer.

10 A factory produces bottles of an energy juice. Two different machines are used to fill empty bottles with the juice. The manager chooses a random sample of 50 bottles filled by machine \(X\) and a random sample of 60 bottles filled by machine \(Y\). The volumes of juice, \(x\) and \(y\) respectively, measured in appropriate units, are summarised by $$\Sigma x = 45.5 , \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 19.56 , \quad \Sigma y = 72.3 , \quad \Sigma ( y - \bar { y } ) ^ { 2 } = 30.25$$ where \(\bar { x }\) and \(\bar { y }\) are the sample means of the volume of juice in the bottles filled by \(X\) and \(Y\) respectively.

1. Find a 90\% confidence interval for the difference between the mean volume of juice in bottles filled by machine \(X\) and the mean volume of juice in bottles filled by machine \(Y\).
   A test at the \(\alpha \%\) significance level does not provide evidence that there is any difference in the means of the volume of juice in bottles filled by machine \(X\) and the volume of juice in bottles filled by machine \(Y\).
2. Find the set of possible values of \(\alpha\).

A large number of people attended a course to improve the speed of their logical thinking. The times taken to complete a particular type of logic puzzle at the beginning of the course and at the end of the course are recorded for each person. The time taken, in minutes, at the beginning of the course is denoted by \(x\) and the time taken, in minutes, at the end of the course is denoted by \(y\). For a random sample of 9 people, the results are summarised as follows. $$\Sigma x = 45.3 \quad \Sigma x ^ { 2 } = 245.59 \quad \Sigma y = 40.5 \quad \Sigma y ^ { 2 } = 195.11 \quad \Sigma x y = 218.72$$ Ken attended the course, but his time to complete the puzzle at the beginning of the course was not recorded. His time to complete the puzzle at the end of the course was 4.2 minutes.

1. By finding, showing all necessary working, the equation of a suitable regression line, find an estimate for the time that Ken would have taken to complete the puzzle at the beginning of the course.
   The values of \(x - y\) for the sample of 9 people are as follows. $$\begin{array} { l l l l l l l l l } 0.2 & 0.8 & 0.5 & 1.0 & 0.2 & 0.6 & 0.2 & 0.5 & 0.8 \end{array}$$ The organiser of the course believes that, on average, the time taken to complete the puzzle decreases between the beginning and the end of the course by more than 0.3 minutes.
2. Stating suitable hypotheses and assuming a normal distribution, test the organiser's belief at the \(2 \frac { 1 } { 2 } \%\) significance level.

A farmer \(A\) grows two types of potato plants, Royal and Majestic. A random sample of 10 Royal plants is taken and the potatoes from each plant are weighed. The total mass of potatoes on a plant is \(x \mathrm {~kg}\). The data are summarised as follows. $$\Sigma x = 42.0 \quad \Sigma x ^ { 2 } = 180.0$$ A random sample of 12 Majestic plants is taken. The total mass of potatoes on a plant is \(y \mathrm {~kg}\). The data are summarised as follows. $$\Sigma y = 57.6 \quad \Sigma y ^ { 2 } = 281.5$$

1. Test, at the \(5 \%\) significance level, whether the population mean mass of potatoes from Royal plants is the same as the population mean mass of potatoes from Majestic plants. You may assume that both distributions are normal and you should state any additional assumption that you make.
   A neighbouring farmer \(B\) grows Crown potato plants. His plants produce 3.8 kg of potatoes per plant, on average. Farmer \(A\) claims that her Royal plants produce a higher mean mass of potatoes than Farmer \(B\) 's Crown plants.
2. Test, at the \(5 \%\) significance level, whether Farmer \(A\) 's claim is justified.

2 In the vegetable section of a local supermarket, leeks are on sale either loose (and unprepared) or prepared in packs of 4 . The weights of unprepared leeks are modelled by the random variable \(X\) which has the Normal distribution with mean 260 grams and standard deviation 24 grams. The prepared leeks have had \(40 \%\) of their weight removed, so that their weights, \(Y\), are modelled by \(Y = 0.6 X\).

1. Find the probability that a randomly chosen unprepared leek weighs less than 300 grams.
2. Find the probability that a randomly chosen prepared leek weighs more than 175 grams.
3. Find the probability that the total weight of 4 randomly chosen prepared leeks in a pack is less than 600 grams.
4. What total weight of prepared leeks in a randomly chosen pack of 4 is exceeded with probability 0.975 ?
5. Sandie is making soup. She uses 3 unprepared leeks and 2 onions. The weights of onions are modelled by the Normal distribution with mean 150 grams and standard deviation 18 grams. Find the probability that the total weight of her ingredients is more than 1000 grams.
6. A large consignment of unprepared leeks is delivered to the supermarket. A random sample of 100 of them is taken. Their weights have sample mean 252.4 grams and sample standard deviation 24.6 grams. Find a \(99 \%\) confidence interval for the true mean weight of the leeks in this consignment.

3 Engineers in charge of a chemical plant need to monitor the temperature inside a reaction chamber. Past experience has shown that when functioning correctly the temperature inside the chamber can be modelled by a Normal distribution with mean \(380 ^ { \circ } \mathrm { C }\). The engineers are concerned that the mean operating temperature may have fallen. They decide to test the mean using the following random sample of 12 recent temperature readings.

1. Give three reasons why a \(t\) test would be appropriate.
2. Carry out the test using a \(5 \%\) significance level. State your hypotheses and conclusion carefully.
3. Find a 95\% confidence interval for the true mean temperature in the reaction chamber.
4. Describe briefly one advantage and one disadvantage of having a 99\% confidence interval instead of a 95\% confidence interval.

A study was made of adult men from region \(A\) of a country. It was found that their heights were normally distributed with a mean of 175.4 cm and standard deviation 6.8 cm .

1. Find the proportion of these men that are taller than 180 cm . A student claimed that the mean height of adult men from region \(B\) of this country was different from the mean height of adult men from region \(A\). A random sample of 52 adult men from region \(B\) had a mean height of 177.2 cm
   The student assumed that the standard deviation of heights of adult men was 6.8 cm both for region \(A\) and region \(B\).
2. Use a suitable test to assess the student's claim. You should
   • state your hypotheses clearly
   • use a \(5 \%\) level of significance
   • Find the \(p\)-value for the test in part (b)

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 \(n\).

4 The distance a particular football player runs in a match is modelled by a normal distribution with standard deviation 0.3 kilometres. A random sample of \(n\) matches is taken.
The distance the player runs in this sample of matches has mean 10.8 kilometres.
The sample is used to construct a \(93 \%\) confidence interval for the mean, of width 0.0543 kilometres, correct to four decimal places. 4

1. Find the value of \(n\) 4
2. Find the \(93 \%\) confidence interval for the mean, giving the limits to three decimal places.
   4
3. Alison claims that the population mean distance the player runs is 10.7 kilometres. She carries out a hypothesis test at the 7\% level of significance using the random sample and the hypotheses $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 10.7 \\ & \mathrm { H } _ { 1 } : \mu \neq 10.7 \end{aligned}$$ 4 (c) (i) State, with a reason, whether the null hypothesis will be accepted or rejected. 4 (c) (ii) Describe, in the context of the hypothesis test in part (c)(i), what is meant by a Type II error. \includegraphics[max width=\textwidth, alt={}, center]{9be40ed6-6df8-426a-8afd-fefc17287de6-06_2488_1730_219_141}

4 The greatest weight \(W N\) that can be supported by a shelving bracket of traditional design is a normally distributed random variable with mean 500 and standard deviation 80 . A sample of 40 shelving brackets of a new design are tested and it is found that the mean of the greatest weights that the brackets in the sample can support is 473.0 N .

1. Test at the \(1 \%\) significance level whether the mean of the greatest weight that a bracket of the new design can support is less than the mean of the greatest weight that a bracket of the traditional design can support.
2. State an assumption needed in carrying out the test in part (a).
3. Explain whether it is necessary to use the central limit theorem in carrying out the test.

6 The random variable \(X\) was assumed to have a normal distribution with mean \(\mu\). Using a random sample of size 128, a significance test was carried out using the following hypotheses. \(\mathrm { H } _ { 0 } : \mu = 30\) \(\mathrm { H } _ { 1 } : \mu > 30\) It was found that \(\sum x = 3929.6\) and \(\sum x ^ { 2 } = 123483.52\). The conclusion of the test was to reject the null hypothesis.

1. Determine the range of possible values of the significance level of the test.
2. It was subsequently found that \(X\) was not normally distributed. Explain whether this invalidates the conclusion of the test.

7 A club secretary collects data about the time, \(T\) minutes, needed to process the details of a new member. The mean of \(T\) is denoted by \(\mu\). The variance of \(T\) is denoted by \(\sigma ^ { 2 }\). The results of a random sample of 40 observations of \(T\) are summarised as follows. \(\mathrm { n } = 40 \quad \Sigma \mathrm { t } = 396.0 \quad \Sigma \mathrm { t } ^ { 2 } = 4271.40\)

1. Determine a 99\% confidence interval for \(\mu\).
2. The secretary discovers that over a long period the value of \(\sigma ^ { 2 }\) is in fact 10.0 . The secretary collects an independent random sample of 50 observations of \(T\) and constructs a new 99\% confidence interval for \(\mu\) based on this sample of size 50 , but using \(\sigma ^ { 2 } = 10.0\). Find the probability that this new confidence interval contains the value \(\mu + 1.6\).

3 Research suggests that the mean reading age of a child about to start secondary school is 10.75 . The reading ages, \(X\) years, of a random sample of 80 children who were about to start secondary school in a particular district were measured, and the results are summarised as follows. $$\mathrm { n } = 80 \quad \sum \mathrm { x } = 893 \quad \sum \mathrm { x } ^ { 2 } = 10267$$

1. Test at the \(5 \%\) significance level whether the mean reading age of children about to start secondary school in this district is not 10.75 .
2. A student wrote: "Although we do not know that the distribution of \(X\) is normal, the central limit theorem allows us to assume that it is, as the sample size is large." This statement is incorrect. Give a corrected version of the student's statement.

4 A random sample of 160 observations of a random variable \(X\) is selected. The sample can be summarised as follows. \(n = 160 \quad \sum x = 2688 \quad \sum x ^ { 2 } = 48398\)

1. Calculate unbiased estimates of the following.
   1. \(\mathrm { E } ( X )\)
   2. \(\operatorname { Var } ( X )\)
2. Find a 99\% confidence interval for \(\mathrm { E } ( X )\), giving the end-points of the interval correct to 4 significant figures.
3. Explain whether it was necessary to use the Central Limit Theorem in answering
   1. part (a),
   2. part (b).