Two-tail z-test

5 Last year the mean time for pizza deliveries from Pete's Pizza Pit was 32.4 minutes. This year the time, \(t\) minutes, for pizza deliveries from Pete's Pizza Pit was recorded for a random sample of 50 deliveries. The results were as follows. $$n = 50 \quad \Sigma t = 1700 \quad \Sigma t ^ { 2 } = 59050$$

1. Find unbiased estimates of the population mean and variance.
2. Test, at the \(2 \%\) significance level, whether the mean delivery time has changed since last year.
3. Under what circumstances would it not be necessary to use the Central Limit Theorem in answering (b)?

4 In the past, the time spent by customers in a certain shop had mean 12.5 minutes and standard deviation 4.2 minutes. Following a change of layout in the shop, the mean time spent in the shop by a random sample of 50 customers is found to be 13.5 minutes.

1. Assuming that the standard deviation remains at 4.2 minutes, test at the \(5 \%\) significance level whether the mean time spent by customers in the shop has changed.
2. Another random sample of 50 customers is chosen and a similar test at the \(5 \%\) significance level is carried out. State the probability of a Type I error.

4 Last year the mean level of a certain pollutant in a river was found to be 0.034 grams per millilitre. This year the levels of pollutant, \(X\) grams per millilitre, were measured at a random sample of 200 locations in the river. The results are summarised below. $$n = 200 \quad \Sigma x = 6.7 \quad \Sigma x ^ { 2 } = 0.2312$$

1. Calculate unbiased estimates of the population mean and variance.
2. Test, at the \(10 \%\) significance level, whether the mean level of pollutant has changed.

1 The heights of a certain species of deer are known to have standard deviation 0.35 m . A zoologist takes a random sample of 150 of these deer and finds that the mean height of the deer in the sample is 1.42 m .

1. Calculate a 96\% confidence interval for the population mean height.
2. Bubay says that \(96 \%\) of deer of this species are likely to have heights that are within this confidence interval. Explain briefly whether Bubay is correct.

5 The lengths, in centimetres, of worms of a certain kind are normally distributed with mean \(\mu\) and standard deviation 2.3. An article in a magazine states that the value of \(\mu\) is 12.7 . A scientist wishes to test whether this value is correct. He measures the lengths, \(x \mathrm {~cm}\), of a random sample of 50 worms of this kind and finds that \(\sum x = 597.1\). He plans to carry out a test, at the \(1 \%\) significance level, of whether the true value of \(\mu\) is different from 12.7 .

1. State, with a reason, whether he should use a one-tailed or a two-tailed test.
2. Carry out the test.

7 Th mean weig 6 bg 6 carrb s is \(\mu \mathrm { klg }\) ams. An is \(\mathbf { p }\) cto wish s to test wh th \(\mathrm { r } \mu = 20\) He weig a ranch sampe 6 tb \(g\) an \(s\) resh ts are sm marised \(s\) fb low \(s\). $$\Sigma x = 430 \quad \Sigma x ^ { 2 } = 0$$ Carryo the test at the to sig fican e lee 1 . If B e th follw ig lin dpg to cm p ete th an wer(s) to ay q stin (s), th q stin \(\mathrm { m } \quad \mathbf { b } \quad \mathrm { r } ( \mathrm { s } )\) ms tb clearlys n n

3 A machine has produced nails over a long period of time, where the length in millimetres was distributed as \(\mathrm { N } ( 22.0,0.19 )\). It is believed that recently the mean length has changed. To test this belief a random sample of 8 nails is taken and the mean length is found to be 21.7 mm . Carry out a hypothesis test at the \(5 \%\) significance level to test whether the population mean has changed, assuming that the variance remains the same.

2 The times taken by students to complete a task are normally distributed with standard deviation 2.4 minutes. A lecturer claims that the mean time is 17.0 minutes. The times taken by a random sample of 5 students were 17.8, 22.4, 16.3, 23.1 and 11.4 minutes. Carry out a hypothesis test at the \(5 \%\) significance level to determine whether the lecturer's claim should be accepted.

4 The weights, \(X\) kilograms, of rabbits in a certain area have population mean \(\mu \mathrm { kg }\). A random sample of 100 rabbits from this area was taken and the weights are summarised by $$\Sigma x = 165 , \quad \Sigma x ^ { 2 } = 276.25 .$$ Test at the \(5 \%\) significance level the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 1.6\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 1.6\).

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.

4 The areas, \(X \mathrm {~cm} ^ { 2 }\), of petals of a certain kind of flower have mean \(\mu \mathrm { cm } ^ { 2 }\). In the past it has been found that \(\mu = 8.9\). Following a change in the climate, a botanist claims that the mean is no longer 8.9. The areas of a random sample of 200 petals from this kind of flower are measured, and the results are summarized by $$\Sigma x = 1850 , \quad \Sigma x ^ { 2 } = 17850 .$$ Test the botanist's claim at the \(2.5 \%\) significance level.

7 Th mean weit 6 bg 6 carrb s is \(\mu \mathrm { k }\) lg ams. An in p cto wish s to test wh th \(\mathrm { r } \mu = 20\) He weits a rand sampe 6 tb g ach s resh ts are sm marised s fb low s. $$\Sigma x = 430 \quad \Sigma x ^ { 2 } = \theta$$ Carryd th test at the \%o sig fican e lee 1. If B e th follw ig lin dpg to cm p ete th an wer(s) to ay q stin (s), th q stin \(\mathrm { m } \quad \mathbf { b } \quad \mathrm { r } ( \mathrm { s } )\) ms tb clearlys n n

1 The time taken for Samuel to drive home from work is distributed with mean 46 minutes. Samuel discovers a different route and decides to test at the \(5 \%\) level whether the mean time has changed. He tries this route on a large number of different days chosen randomly and calculates the mean time.

1. State the null and alternative hypotheses for this test.
2. Samuel calculates the value of his test statistic \(z\) to be - 1.729 . What conclusion can he draw?

2 The times taken for the pupils in Ming's year group to do their English homework have a normal distribution with standard deviation 15.7 minutes. A teacher estimates that the mean time is 42 minutes. The times taken by a random sample of 3 students from the year group were 27, 35 and 43 minutes. Carry out a hypothesis test at the \(10 \%\) significance level to determine whether the teacher's estimate for the mean should be accepted, stating the null and alternative hypotheses.

6 A clinic monitors the amount, \(X\) milligrams per litre, of a certain chemical in the blood stream of patients. For patients who are taking drug \(A\), it has been found that the mean value of \(X\) is 0.336 . A random sample of 100 patients taking a new drug, \(B\), was selected and the values of \(X\) were found. The results are summarised below. $$n = 100 , \quad \Sigma x = 43.5 , \quad \Sigma x ^ { 2 } = 31.56 .$$

1. Test at the \(1 \%\) significance level whether the mean amount of the chemical in the blood stream of patients taking drug \(B\) is different from that of patients taking drug \(A\).
2. For the test to be valid, is it necessary to assume a normal distribution for the amount of chemical in the blood stream of patients taking drug \(B\) ? Justify your answer.

3 Following a change in flight schedules, an airline pilot wished to test whether the mean distance that he flies in a week has changed. He noted the distances, \(x \mathrm {~km}\), that he flew in 50 randomly chosen weeks and summarised the results as follows. $$n = 50 \quad \Sigma x = 143300 \quad \Sigma x ^ { 2 } = 410900000$$

1. Calculate unbiased estimates of the population mean and variance.
2. In the past, the mean distance that he flew in a week was 2850 km . Test, at the \(5 \%\) significance level, whether the mean distance has changed.

1 A researcher wishes to investigate whether the mean height of a certain type of plant in one region is different from the mean height of this type of plant everywhere else. He takes a large random sample of plants from the region and finds the sample mean. He calculates the value of the test statistic, \(z\), and finds that \(z = 1.91\).

1. Explain briefly why the researcher should use a two-tail test.
2. Carry out the test at the \(4 \%\) significance level.

7 An examination board is developing a new syllabus and wants to know if the question papers are the right length. A random sample of 50 candidates was given a pre-test on a dummy paper. The times, \(t\) minutes, taken by these candidates to complete the paper can be summarised by \(n = 50\), \(\sum \boldsymbol { t } = \mathbf { 4 0 5 0 }\), \(\sum \boldsymbol { t } ^ { \mathbf { 2 } } \boldsymbol { = } \mathbf { 3 2 9 8 0 0 }\).
Assume that times are normally distributed.
[0pt]

1. Estimate the proportion of candidates that could not complete the paper within 90 minutes. [6]
2. Test, at the \(10 \%\) significance level, whether the mean time for all candidates to complete this paper is \(\mathbf { 8 0 }\) minutes. Use a two-tail test. [7]
3. Explain whether the assumption that times are normally distributed is necessary in answering
   (a) part (i),
   [0pt] (b) part (ii). [2]

6 Records for a doctors' surgery over a long period suggest that the time taken for a consultation, \(T\) minutes, has a mean of 11.0. Following the introduction of new regulations, a doctor believes that the average time has changed. She finds that, with new regulations, the consultation times for a random sample of 120 patients can be summarised as $$n = 120 , \Sigma t = 1411.20 , \Sigma t ^ { 2 } = 18737.712 .$$

1. Test, at the \(10 \%\) significance level, whether the doctor's belief is correct.
2. Explain whether, in answering part (i), it was necessary to assume that the consultation times were normally distributed.

8 In a large company the time taken for an employee to carry out a certain task is a normally distributed random variable with mean 78.0 s and unknown variance. A new training scheme is introduced and after its introduction the times taken by a random sample of 120 employees are recorded. The mean time for the sample is 76.4 s and an unbiased estimate of the population variance is \(68.9 \mathrm {~s} ^ { 2 }\).

1. Test, at the \(1 \%\) significance level, whether the mean time taken for the task has changed.
2. It is required to redesign the test so that the probability of making a Type I error is less than 0.01 when the sample mean is 77.0 s . Calculate an estimate of the smallest sample size needed, and explain why your answer is only an estimate.

7 A machine is designed to make paper with mean thickness 56.80 micrometres. The thicknesses, \(x\) micrometres, of a random sample of 300 sheets are summarised by $$n = 300 , \quad \Sigma x = 17085.0 , \quad \Sigma x ^ { 2 } = 973847.0 .$$ Test, at the \(10 \%\) significance level, whether the machine is producing paper of the designed thickness.

6 Records show that before the year 1990 the maximum daily temperature \(T ^ { \circ } \mathrm { C }\) at a seaside resort in August can be modelled by a distribution with mean 24.3. The maximum temperatures of a random sample of 50 August days since 1990 can be summarised by $$n = 50 , \quad \Sigma t = 1314.0 , \quad \Sigma t ^ { 2 } = 36602.17 .$$

1. Test, at the \(1 \%\) significance level, whether there is evidence of a change in the mean maximum daily temperature in August since 1990.
2. Give a reason why it is possible to use the Central Limit Theorem in your test.

5 The acidity \(A\) (measured in pH ) of soil of a particular type has a normal distribution. The pH values of a random sample of 80 soil samples from a certain region can be summarised as $$\Sigma a = 496 , \quad \Sigma a ^ { 2 } = 3126 .$$ Test, at the \(10 \%\) significance level, whether in this region the mean pH of soil is 6.1 .

6 The random variable \(X\) denotes the yield, in kilograms per acre, of a certain crop. Under the standard treatment it is known that \(\mathrm { E } ( X ) = 38.4\). Under a new treatment, the yields of 50 randomly chosen regions can be summarised as $$n = 50 , \quad \sum x = 1834.0 , \quad \sum x ^ { 2 } = 70027.37 .$$ Test at the \(1 \%\) level whether there has been a change in the mean crop yield.