5.05c Hypothesis test: normal distribution for population mean

A machine fills bags with flour. The weight of flour delivered by the machine into a bag, \(X\) grams, is normally distributed with mean \(\mu\) grams and standard deviation 30 grams. To check if there is any change to the mean weight of flour delivered by the machine into each bag, Olaf takes a random sample of 10 bags. The weight of flour, \(x\) grams, in each bag is recorded and \(\bar { x } = 1020\)

1. Test, at the \(5 \%\) level of significance, \(\mathrm { H } _ { 0 } : \mu = 1000\) against \(\mathrm { H } _ { 1 } : \mu \neq 1000\) Olaf decides to alter the test so that the hypotheses are \(\mathrm { H } _ { 0 } : \mu = 1000\) and \(\mathrm { H } _ { 1 } : \mu > 1000\) but keeps the level of significance at 5\% He takes a second sample of size \(n\) and finds the critical region, \(\bar { X } > c\)
2. Find an equation for \(c\) in terms of \(n\) When the true value of \(\mu\) is 1020 grams, the probability of making a Type II error is 0.0050 , to 2 significant figures.
3. Calculate the value of \(n\) and the value of \(c\)

1. A machine fills cartons with juice.

The amount of juice in a carton is normally distributed with mean \(\mu \mathrm { ml }\) and standard deviation 8 ml . A manager wants to test whether or not the amount of juice in the cartons, \(X \mathrm { ml }\), is less than 330 ml . The manager takes a random sample of 25 cartons of juice and calculates the mean amount of juice \(\bar { x } \mathrm { ml }\).

1. Using a \(5 \%\) level of significance, find the critical region of \(\bar { X }\) for this test. State your hypotheses clearly. The Director is concerned about the machine filling the cartons with more than 330 ml of juice as well as less than 330 ml of juice. The Director takes a sample of 55 cartons, records the mean amount of juice \(\bar { y } \mathrm { ml }\) and uses a test with a critical region of $$\{ \bar { Y } < 328 \} \cup \{ \bar { Y } > 332 \}$$
2. Find P (Type I error) for the Director's test. When \(\mu = 325 \mathrm { ml }\)
3. find P (Type II error) for the test in part (a)

1. The number of errors made by a secretary is modelled by a Poisson distribution with a mean of 2.4 per 100 words.

A 100-word piece of work completed by the secretary is selected at random.

1. Find the probability that
   1. there are exactly 3 errors,
   2. there are fewer than 2 errors. After a long holiday, a randomly selected piece of work containing 250 words completed by the secretary is examined to see if the rate of errors has changed.
2. Stating your hypotheses clearly, and using a \(5 \%\) level of significance, find the critical region for a suitable test.
3. Find P (Type I error) for the test in part (b)

Every morning Geethaka repeatedly rolls a fair, six-sided die until he rolls a 3 and then he stops. The random variable \(X\) represents the number of times he rolls the die each morning.

1. Suggest a suitable model for the random variable \(X\)
2. Show that \(\mathrm { P } ( X \leqslant 3 ) = \frac { 91 } { 216 }\) After 64 mornings Geethaka will calculate the mean number of times he rolled the die.
3. Estimate the probability that the mean number of rolls is between 5.6 and 7.2 Nira wants to check Geethaka's die to decide whether or not the probability of rolling a 3 with his die is less than \(\frac { 1 } { 6 }\) Nira rolls the die repeatedly until she rolls a 3
   She obtains \(x = 16\)
4. By carrying out a suitable test, determine what Nira's conclusion should be. You should state your hypotheses clearly and use a \(5 \%\) level of significance.

1. Bacteria are randomly distributed in a river at a rate of 5 per litre of water. A new factory opens and a scientist claims it is polluting the river with bacteria. He takes a sample of 0.5 litres of water from the river near the factory and finds that it contains 7 bacteria. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether there is evidence that the level of pollution has increased.

\section*{Q uestion 1 continued}

1. Sam and Tessa are testing a spinner to see if the probability, p , of it landing on red is less than \(\frac { 1 } { 5 }\). They both use a \(10 \%\) significance level.

Sam decides to spin the spinner 20 times and record the number of times it lands on red.

1. Find the critical region for Sam's test.
2. Write down the size of Sam's test. Tessa decides to spin the spinner until it lands on red and she records the number of spins.
3. Find the critical region for Tessa's test.
4. Find the size of Tessa's test.
5. 1. Show that the power function for Sam's test is given by $$( 1 - p ) ^ { 19 } ( 1 + 19 p )$$
   2. Find the power function for Tessa's test.
6. With reference to parts (b), (d) and (e), state, giving your reasons, whether you would recommend Sam's test or Tessa's test when \(\mathrm { p } = 0.15\)

3 Yin grows two varieties of potato, plant \(A\) and plant \(B\). A random sample of each variety of potato is taken and the yield, \(x \mathrm {~kg}\), produced by each plant is measured. The following statistics are obtained from the data.

1. Stating your hypotheses clearly, test, at the \(10 \%\) significance level, whether or not the variances of the yields of the two varieties of potato are the same.
2. State an assumption you have made in order to carry out the test in part (a).

6 A company manufactures bolts. The diameter of the bolts follows a normal distribution with a mean diameter of 5 mm . Stan believes that the mean diameter of the bolts is less than 5 mm . He takes a random sample of 10 bolts and measures their diameters. He calculates some statistics but spills ink on his work before completing them. The only information he has left is as follows \includegraphics[max width=\textwidth, alt={}, center]{67df73d4-6ce4-45f7-8a69-aa94292ea814-16_394_1150_527_456} Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not Stan's belief is supported.

1 Gina receives a large number of packages from two companies, \(A\) and \(B\). She believes that the variance of the weights of packages from company \(A\) is greater than the variance of the weights of packages from company \(B\). Gina takes a random sample of 7 packages from company \(A\) and an independent random sample of 10 packages from company \(B\). Her results are summarised below $$\bar { a } = 300 \quad \mathrm {~S} _ { a a } = 145496 \quad \bar { b } = 233.4 \quad \mathrm {~S} _ { b b } = 56364.4$$ [You may assume that the weights of packages from the two companies are normally distributed.]
Test Gina's belief. Use a \(5 \%\) level of significance and state your hypotheses clearly.

6 A new employee, Kim, joins an existing employee, Jiang, to work in the quality control department of a company producing steel rods.
Each day a random sample of rods is taken, their lengths measured and a \(95 \%\) confidence interval for the mean length of the rods, in metres, is calculated. It is assumed that the lengths of the rods produced are normally distributed. Kim took a random sample of 25 rods and used the \(t\) distribution to obtain a \(95 \%\) confidence interval of \(( 1.193,1.367 )\) for the mean length of the rods. Jiang commented that this interval was a little wider than usual and explained that they usually assume that the standard deviation does not change and can be taken as 0.175 metres.

1. Test, at the \(10 \%\) level of significance, whether or not Kim's sample suggests that the standard deviation is different from 0.175 metres. State your hypotheses clearly. Using Kim's sample and the normal distribution with a standard deviation of 0.175 metres, (b) find a 95\% confidence interval for the mean length of the rods.

1. A company produces two colours of candles, blue and white. The standard deviation of the burning times of the blue candles is 2.6 minutes and the standard deviation of the burning times of the white candles is 2.4 minutes.

Nissim claims that the mean burning time of blue candles is more than 5 minutes greater than the mean burning time of white candles. A random sample of 90 blue candles is found to have a mean burning time of 39.5 minutes. A random sample of 80 white candles is found to have a mean burning time of 33.7 minutes.

1. Stating your hypotheses clearly, use a suitable test to assess Nissim's belief. Use a \(1 \%\) level of significance.
2. Explain how the hypothesis test in part (a) would be carried out differently if the variances of the burning times of candles were unknown. The burning times for the candles may not follow a normal distribution.
3. Describe the effect this would have on the calculations in the hypothesis test in part (a). Give a reason for your answer.

1. Elsa is collecting information on the wingspan of two different species of butterfly, Ringlet and Meadow Brown. She takes a random sample of each type of butterfly. The wingspans, \(w \mathrm {~cm}\), are summarised in the table below. The wingspans of Ringlet and Meadow Brown butterflies each follow normal distributions.

1. Test, at the \(2 \%\) level of significance, whether or not there is evidence that the variance of the wingspans of Ringlet butterflies is different from the variance of the wingspans of Meadow Brown butterflies. You should state your hypotheses clearly. The \(k \%\) confidence interval for the variance of the wingspans of Meadow Brown butterflies is (1.194, 48.54)
2. Find the value of \(k\)
3. Calculate a \(95 \%\) confidence interval for the difference between the mean wingspan of the Ringlet butterfly and the mean wingspan of the Meadow Brown butterfly.

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. A doctor believes that a four-week exercise programme can reduce the resting heart rate of her patients. She takes a random sample of 7 patients and records their resting heart rate before the exercise programme and again after the exercise programme.

1. Using a \(5 \%\) level of significance, carry out an appropriate test of the doctor's belief. You should state your hypotheses, test statistic and critical value.
2. State the assumption made about the resting heart rates that was required to carry out the test.

1. The concentration of an air pollutant is measured in micrograms \(/ \mathrm { m } ^ { 3 }\)

Samples of air were taken at two different sites and the concentration of this particular air pollutant was recorded. For Site \(A\) the summary statistics are shown below. For Site \(B\) there were 9 samples of air taken.
A test of the hypothesis \(\mathrm { H } _ { 0 } : \sigma _ { A } ^ { 2 } = \sigma _ { B } ^ { 2 }\) against the hypothesis \(\mathrm { H } _ { 1 } : \sigma _ { A } ^ { 2 } \neq \sigma _ { B } ^ { 2 }\) is carried out using a \(2 \%\) level of significance.

1. State a necessary assumption required to carry out the test. Given that the assumption in part (a) holds,
2. find the set of values of \(s _ { B } ^ { 2 }\) that would lead to the null hypothesis being rejected,
3. find a 99\% confidence interval for the variance of the concentration of the air pollutant at Site A.

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. Two machines, \(A\) and \(B\), are used to fill bottles of water. The amount of water dispensed by each machine is normally distributed.

Samples are taken from each machine and the amount of water, \(x \mathrm { ml }\), dispensed in each bottle is recorded. The table shows the summary statistics for Machine \(A\).

1. Find a 95\% confidence interval for the variance of the amount of water dispensed in each bottle by Machine \(A\). For Machine \(B\), a random sample of 11 bottles is taken. The sample variance of the amount of water dispensed in bottles is \(12.7 \mathrm { ml } ^ { 2 }\)
2. Test, at the \(10 \%\) level of significance, whether there is evidence that the variances of the amounts of water dispensed in bottles by the two machines are different. You should state the hypotheses and the critical value used.

1. A psychologist claims to have developed a technique to improve a person's memory.

A random sample of 8 people are each given the same list of words to memorise and recall. Each person then receives memory training from the psychologist. After the training, each person is given the same list of new words to memorise and recall. The table shows the percentage of words recalled by each person before and after the training.

1. State why a paired \(t\)-test is suitable for these data.
2. State an assumption that needs to be made in order to carry out a paired \(t\)-test in this case.
3. Test, at the \(5 \%\) level of significance, whether or not there is evidence of an increase in the percentage of words recalled after receiving the psychologist's training. State your hypotheses, test statistic and critical value used for this 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 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.

1. A researcher set up a trial to assess the effect that a food supplement has on the increase in weight of Herdwick lambs. The researcher randomly selected 8 sets of twin lambs. One of each set of twins was given the food supplement and the other had no food supplement. The gain in weight, in kg, of each lamb over the period of the trial was recorded.

1. State why a two sample \(t\)-test is not suitable for use with these data.
2. Suggest 2 other factors about the lambs that the researcher may need to control when selecting the sample.
3. State one assumption, in context, that needs to be made for a paired \(t\)-test to be valid. For a pair of twin lambs, the random variable \(W\) represents the weight gain of the lamb given the food supplement minus the weight gain of the lamb not given the food supplement.
4. Using the data in the table, calculate a \(98 \%\) confidence interval for the mean of \(W\) Show your working clearly. The researcher believes that the mean of \(W\) is greater than 200 g
5. Stating your hypotheses clearly, use your confidence interval to explain whether or not there is evidence to support the researcher's belief.

1. A nutritionist studied the levels of cholesterol, \(X \mathrm { mg } / \mathrm { cm } ^ { 3 }\), of male students at a large college. She assumed that \(X\) was distributed \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\) and examined a random sample of 25 male students. Using this sample she obtained unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\) as \(\hat { \mu }\) and \(\hat { \sigma } ^ { 2 }\)

A \(95 \%\) confidence interval for \(\mu\) was found to be \(( 1.128,2.232 )\)

1. Show that \(\hat { \sigma } ^ { 2 } = 1.79\) (correct to 3 significant figures)
2. Obtain a \(95 \%\) confidence interval for \(\sigma ^ { 2 }\)

1. The times, \(x\) seconds, taken by the competitors in the 100 m freestyle events at a school swimming gala are recorded. The following statistics are obtained from the data.

Following the gala, a mother claims that girls are faster swimmers than boys. Assuming that the times taken by the competitors are two independent random samples from normal distributions,

1. 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 the mother's claim. Use a \(5 \%\) level of significance.

8 In this question you must show detailed reasoning. On the manufacturer's website, it is claimed that the average daily electricity consumption of a particular model of fridge is 1.25 kWh (kilowatt hours). A researcher at a consumer organisation decides to check this figure. A random sample of 40 fridges is selected. Summary statistics for the electricity consumption \(x \mathrm { kWh }\) of these fridges, measured over a period of 24 hours, are as follows. \(\Sigma x = 51.92 \quad \Sigma x ^ { 2 } = 70.57\) Carry out a test at the \(5 \%\) significance level to investigate the validity of the claim on the website.
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6 The reaction times, in milliseconds, of all adult males in a standard experiment have a symmetrical distribution with mean and median both equal to 700 and standard deviation 125. The reaction times of a random sample of 6 international athletes are measured and the results are as follows: \(\begin{array} { l l l l l l } 702 & 631 & 540 & 714 & 575 & 480 \end{array}\) It is required to test whether international athletes have a mean reaction time which is less than 700.

1. Assume first that the reaction times of international athletes have the distribution \(\mathrm { N } \left( \mu , 125 ^ { 2 } \right)\). Test at the \(5 \%\) significance level whether \(\mu < 700\).
2. Now assume only that the distribution of the data is symmetrical, but not necessarily normal.
   1. State with a reason why a Wilcoxon test is preferable to a sign test.
   2. Use an appropriate Wilcoxon test at the \(5 \%\) significance level to test whether the median reaction time of international athletes is less than 700 .
3. Explain why the significance tests in part (a) and part (b)(ii) could produce different results.