5.05b Unbiased estimates: of population mean and variance

4 The number of accidents per 3-month period on a certain road has the distribution \(\operatorname { Po } ( \lambda )\). In the past the value of \(\lambda\) has been 5.7. Following some changes to the road, the council carries out a hypothesis test to determine whether the value of \(\lambda\) has decreased. If there are fewer than 3 accidents in a randomly chosen 3 -month period, the council will conclude that the value of \(\lambda\) has decreased.

1. Find the probability of a Type I error.
2. Find the probability of a Type II error if the mean number of accidents per 3-month period is now actually 0.9 .

1 The lengths, \(X\) centimetres, of a random sample of 7 leaves from a certain variety of tree are as follows.
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1. Calculate unbiased estimates of the population mean and variance of \(X\).
   It is now given that the true value of the population variance of \(X\) is 0.55 , and that \(X\) has a normal distribution.
2. Find a 95\% confidence interval for the population mean of \(X\).

2 A shop obtains apples from a certain farm. It has been found that 5\% of apples from this farm are Grade A. Following a change in growing conditions at the farm, the shop management plan to carry out a hypothesis test to find out whether the proportion of Grade A apples has increased. They select 25 apples at random. If the number of Grade A apples is more than 3 they will conclude that the proportion has increased.

1. State suitable null and alternative hypotheses for the test.
2. Find the probability of a Type I error.
   In fact 2 of the 25 apples were Grade A .
3. Which of the errors, Type I or Type II, is possible? Justify your answer.

4 The mean time to mark a certain set of examination papers is estimated by the examination board to be 12 minutes per paper. A random sample of 150 examination papers gave \(\Sigma x = 2130\) and \(\Sigma x ^ { 2 } = 37746\), where \(x\) is the time in minutes to mark an examination paper.

1. Calculate unbiased estimates of the population mean and variance.
2. Stating the null and alternative hypotheses, use a \(10 \%\) significance level to test whether the examination board's estimated time is consistent with the data.

5 To test whether a coin is biased or not, it is tossed 10 times. The coin will be considered biased if there are 9 or 10 heads, or 9 or 10 tails.

1. Show that the probability of making a Type I error in this test is approximately 0.0215 .
2. Find the probability of making a Type II error in this test when the probability of a head is actually 0.7.

2 Before attending a basketball course, a player found that \(60 \%\) of his shots made a score. After attending the course the player claimed he had improved. In his next game he tried 12 shots and scored in 10 of them. Assuming shots to be independent, test this claim at the \(10 \%\) significance level.

5 Over a long period of time it is found that the time spent at cash withdrawal points follows a normal distribution with mean 2.1 minutes and standard deviation 0.9 minutes. A new system is tried out, to speed up the procedure. The null hypothesis is that the mean time spent is the same under the new system as previously. It is decided to reject the null hypothesis and accept that the new system is quicker if the mean withdrawal time from a random sample of 20 cash withdrawals is less than 1.7 minutes. Assume that, for the new system, the standard deviation is still 0.9 minutes, and the time spent still follows a normal distribution.

1. Calculate the probability of a Type I error.
2. If the mean withdrawal time under the new system is actually 1.5 minutes, calculate the probability of a Type II error.

1 A random sample of 100 values of a variable \(X\) is taken. These values are summarised below. $$n = 100 \quad \Sigma x = 1556 \quad \Sigma x ^ { 2 } = 29004$$ Calculate unbiased estimates of the population mean and variance of \(X\).

6 The heights, \(h\) centimetres, of a random sample of 100 fully grown animals of a certain species were measured. The results are summarised below. $$n = 100 \quad \Sigma h = 7570 \quad \Sigma h ^ { 2 } = 588050$$

1. Find unbiased estimates of the population mean and variance.
2. Calculate a \(99 \%\) confidence interval for the mean height of animals of this species.
   Four random samples were taken and a \(99 \%\) confidence interval for the population mean, \(\mu\), was found from each sample.
3. Find the probability that all four of these confidence intervals contain the true value of \(\mu\).

4 The masses, \(m\) kilograms, of flour in a random sample of 90 sacks of flour are summarised as follows. $$n = 90 \quad \Sigma m = 4509 \quad \Sigma m ^ { 2 } = 225950$$

1. Find unbiased estimates of the population mean and variance.
2. Calculate a \(98 \%\) confidence interval for the population mean.
3. Explain why it was necessary to use the Central Limit theorem in answering part (b).
4. Find the probability that the confidence interval found in part (b) is wholly above the true value of the population mean.

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1. A javelin thrower noted the lengths of a random sample of 50 of her throws. The sample mean was 72.3 m and an unbiased estimate of the population variance was \(64.3 \mathrm {~m} ^ { 2 }\). Find a \(92 \%\) confidence interval for the population mean length of throws by this athlete.
2. A discus thrower wishes to calculate a \(92 \%\) confidence interval for the population mean length of his throws. He bases his calculation on his first 50 throws in a week. Comment on this method.

3 Batteries of type \(A\) are known to have a mean life of 150 hours. It is required to test whether a new type of battery, type \(B\), has a shorter mean life than type \(A\) batteries.

1. Give a reason for using a sample rather than the whole population in carrying out this test.
   A random sample of 120 type \(B\) batteries are tested and it is found that their mean life is 147 hours, and an unbiased estimate of the population variance is 225 hours \(^ { 2 }\).
2. Test, at the \(2 \%\) significance level, whether type \(B\) batteries have a shorter mean life than type \(A\) batteries.
3. Calculate a \(94 \%\) confidence interval for the population mean life of type \(B\) batteries.

6 A random sample of 5 values of a variable \(X\) is given below. $$\begin{array} { l l l l l } 2 & 3 & 3 & 5 & a \end{array}$$

1. Find an expression, in terms of \(a\), for the mean of these values.
   It is given that an unbiased estimate of the population variance of \(X\), using these values, is 4 . It is also given that \(a\) is positive.
2. Find and simplify a quadratic equation in terms of \(a\) and hence find the value of \(a\).

6 A sample of 5 randomly selected values of a variable \(X\) is as follows: $$\begin{array} { l l l l l } 1 & 2 & 6 & 1 & a \end{array}$$ where \(a > 0\).
Given that an unbiased estimate of the variance of \(X\) calculated from this sample is \(\frac { 11 } { 2 }\), find the value of \(a\).

3 The masses, in kilograms, of newborn babies in country \(A\) are represented by the random variable \(X\), with mean \(\mu\) and variance \(\sigma ^ { 2 }\). The masses of a random sample of 500 newborn babies in this country were found and the results are summarised below. $$n = 500 \quad \Sigma x = 1625 \quad \Sigma x ^ { 2 } = 5663.5$$

1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
   A researcher wishes to test whether the mean mass of newborn babies in a neighbouring country, \(B\), is different from that in country \(A\). He chooses a random sample of 60 newborn babies in country \(B\) and finds that their sample mean mass is 2.95 kg . Assume that your unbiased estimates in part (a) are the correct values for \(\mu\) and \(\sigma ^ { 2 }\). Assume also that the variance of the masses of newborn babies in country \(B\) is the same as in country \(A\).
2. Carry out the test at the \(1 \%\) significance level.

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)?

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1. A random sample of 8 boxes of cereal from a certain supplier was taken. Each box was weighed and the masses in grams were as follows. $$\begin{array} { l l l l l l l l } 261 & 249 & 259 & 252 & 255 & 256 & 258 & 254 \end{array}$$ Find unbiased estimates of the population mean and variance.
2. The supplier claims that the mean mass of boxes of cereal is 253 g . A quality control officer suspects that the mean mass is actually more than 253 g . In order to test this claim, he weighs a random sample of 100 boxes of cereal and finds that the total mass is 25360 g .
   1. Given that the population standard deviation of the masses is 3.5 g , test at the \(5 \%\) significance level whether the population mean mass is more than 253 g .
      An employee says, 'This test is invalid because it uses the normal distribution, but we do not know whether the masses of the boxes are normally distributed.'
   2. Explain briefly whether this statement is true or not.

7 Every July, as part of a research project, Rita collects data about sightings of a particular kind of bird. Each day in July she notes whether she sees this kind of bird or not, and she records the number \(X\) of days on which she sees it. She models the distribution of \(X\) by \(\mathrm { B } ( 31 , p )\), where \(p\) is the probability of seeing this kind of bird on a randomly chosen day in July. Data from previous years suggests that \(p = 0.3\), but in 2022 Rita suspected that the value of \(p\) had been reduced. She decided to carry out a hypothesis test. In July 2022, she saw this kind of bird on 4 days.

1. Use the binomial distribution to test at the \(5 \%\) significance level whether Rita's suspicion is justified.
   In July 2023, she noted the value of \(X\) and carried out another test at the \(5 \%\) significance level using the same hypotheses.
2. Calculate the probability of a Type I error.
   Rita models the number of sightings, \(Y\), per year of a different, very rare, kind of bird by the distribution \(B ( 365,0.01 )\).
3. 1. Use a suitable approximating distribution to find \(\mathrm { P } ( Y = 4 )\).
   2. Justify your approximating distribution in this context.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 Henri wants to choose a random sample from the 804 students at his college. He numbers the students from 1 to 804 and then uses random numbers generated by his calculator. The first 20 random digits produced by his calculator are as follows. $$\begin{array} { l l l l l l l l l l l l l l l l l l l l } 5 & 6 & 7 & 1 & 0 & 9 & 8 & 4 & 3 & 1 & 0 & 9 & 6 & 6 & 5 & 0 & 2 & 1 & 7 & 6 \end{array}$$ Henri's first two student numbers are 567 and 109.

1. Use Henri's digits to find the numbers of the next two students in the sample.
   There were 30 students in Henri's sample. He asked each of them how much time, \(X\) hours, they spent on social media each week, on average. He summarised the results as follows. $$n = 30 \quad \Sigma x = 610 \quad \Sigma x ^ { 2 } = 12405$$
2. Use this information to calculate an unbiased estimate of the mean of \(X\) and show that an unbiased estimate of the variance of \(X\) is less than 0.1 .
3. Henri's friend claims that Henri has probably made a mistake in his calculation of \(\Sigma x\) or \(\Sigma x ^ { 2 }\). Use your answer to part (b) to comment on this claim.

4 In this question you should not use an approximating distribution.
At an election in Menham last year, \(24 \%\) of voters supported the Today Party. A student wishes to test whether support for the Today Party has decreased since last year. He chooses a random sample of 25 voters in Menham and finds that exactly 2 of them say that they support the Today Party. Test at the 5\% significance level whether support for the Today Party has decreased.

6 The numbers of green sweets in 200 randomly chosen packets of Frutos are summarised in the table.

1. Calculate an unbiased estimate for the population mean of the number of green sweets in a packet of Frutos, and show that an unbiased estimate of the population variance is 0.783 correct to 3 significant figures.
   The manufacturers of Frutos claim that the mean number of green sweets in a packet is 1.65 .
   Anji believes that the true value of the mean, \(\mu\), is less than 1.65 . She uses the results from the 200 randomly chosen packets to test the manufacturers' claim.
2. State suitable null and alternative hypotheses for the test. \includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-08_2714_37_143_2008}
3. Show that the result of Anji's test is significant at the \(5 \%\) level but not at the \(1 \%\) level.
4. It is given that Anji made a Type I error. Explain how this shows that the significance level that Anji used in her test was not \(1 \%\).

3 In the past, the mean time taken by Freda for a particular daily journey was 39.2 minutes. Following the introduction of a one-way system, Freda wishes to test whether the mean time for the journey has decreased. She notes the times, \(t\) minutes, for 40 randomly chosen journeys and summarises the results as follows. $$n = 40 \quad \Sigma t = 1504 \quad \Sigma t ^ { 2 } = 57760$$

1. Calculate unbiased estimates of the population mean and variance of the new journey time.
2. Test, at the \(5 \%\) significance level, whether the population mean time has decreased.

7 A national survey shows that \(95 \%\) of year 12 students use social media. Arvin suspects that the percentage of year 12 students at his college who use social media is less than the national percentage. He chooses a random sample of 20 students at his college and notes the number who use social media. He then carries out a test at the \(2 \%\) significance level.

1. Find the rejection region for the test.
2. Find the probability of a Type I error.
3. Jimmy believes that the true percentage at Arvin's college is \(70 \%\). Assuming that Jimmy is correct, find the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 It is known that \(8 \%\) of adults in a certain town own a Chantor car. After an advertising campaign, a car dealer wishes to investigate whether this proportion has increased. He chooses a random sample of 25 adults from the town and notes how many of them own a Chantor car.

1. He finds that 4 of the 25 adults own a Chantor car. Carry out a hypothesis test at the 5\% significance level.
2. Explain which of the errors, Type I or Type II, might have been made in carrying out the test in part (a).
   Later, the car dealer takes another random sample of 25 adults from the town and carries out a similar hypothesis test at the 5\% significance level.
3. Find the probability of a Type I error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1 The lengths, in millimetres, of a random sample of 12 rods made by a certain machine are as follows.
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1. Find unbiased estimates of the population mean and variance.
2. Give a statistical reason why these estimates may not be reliable.