A LevelMaths: Probability & Statistics 2CIEExam Questions1. Statistical Sampling1.2 Sampling & Estimation

Sampling & Estimation (CIE A Level Maths: Probability & Statistics 2)

Exam Questions

3 hours28 questions
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1a
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Junaid, a teaching assistant at a university, takes a sample of his students’ essays and records the number of spelling mistakes, x, each essay contains. The results are summarised as follows.

n equals 50 space space space space space space space space space space space space space space space space sum x equals 5173 space space space space space space space space space space space space space space space space space sum x squared equals 560024

Use the formula below to calculate an unbiased estimate for the population mean number of spelling mistakes in a student’s essay.

̅ x equals fraction numerator sum x over denominator n end fraction

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1b
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Use the formula below to calculate an unbiased estimate for the population variance of the number of spelling mistakes in a student’s essay.

s to the power of 2 space end exponent equals fraction numerator 1 over denominator n minus 1 end fraction space left parenthesis sum x squared minus fraction numerator left parenthesis sum x right parenthesis squared over denominator n end fraction right parenthesis

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2a
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Mike takes a sample of five snails from a local field and records how far they travel in 10 minutes. The distances, in metres, are shown below.

2.6      3.1      2.9      2.4      3.6

Use the formula below to calculate an unbiased estimate for the population mean.

̅ x space equals space fraction numerator sum x over denominator n end fraction

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2b
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Use the formula below to calculate an unbiased estimate for the population variance.

s to the power of 2 space end exponent equals fraction numerator 1 over denominator n minus 1 end fraction space left parenthesis sum x squared minus fraction numerator left parenthesis sum x right parenthesis squared over denominator n end fraction right parenthesis

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3
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n observations of a random variable X are used to form a random sample. The sample mean is denoted ̅ X. If X has mean mu and variance sigma squared then ̅ X has mean mu and variance sigma squared over n.

Find the E space open parentheses ̅ X close parentheses and V a r space open parentheses ̅ X close parentheses in the case when:

(i)
E space left parenthesis X right parenthesis equals 20 comma space space space space space space space space space space space V a r space left parenthesis X right parenthesis equals 8 space space space space space space space space space a n d space space space space space space space n equals 4
(ii)
E space left parenthesis X right parenthesis equals 50 comma space space space space space space space space space space V a r space left parenthesis X right parenthesis equals 105 space space space space space space a n d space space space space space space n equals 20
(iii)
E space left parenthesis X right parenthesis equals negative 17 comma space space space space space space V a r space left parenthesis X right parenthesis equals 42 space space space space space space space space a n d space space space space space space n equals 30

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4a
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A random sample of n observations of X space tilde space N left parenthesis 10 comma 16 right parenthesis are taken and the distribution of the sample mean is denoted ̅ X subscript n. It is known that the sample mean also follows a normal distribution regardless of the size of the sample.

(i)
In the case where the sample size is 25, write down the distribution of the sample mean, ̅ X subscript 25.
(ii)
Write down the standard deviation of ̅ X subscript 25.
(iii)
Find P left parenthesis 9 less than ̅ X subscript 25 less than 11 right parenthesis.

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4b
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State which distribution, ̅ X subscript 20 or ̅ X subscript 50, will have the smallest standard deviation. Give a reason for your answer.

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4c
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Given that the variance of ̅ X subscript n is 0.16, state the value of n.

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5
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For each of the following scenarios, state with a reason whether the Central Limit theorem (CLT) is needed for the random variable, ̅ X, to be modelled as a normal distribution.

(i)
The lengths of unicorn horns are known to be normally distributed. A random sample of 20 unicorns are taken and the mean length of their horns is calculated, ̅ X .
(ii)
In an experiment, each participant is asked to flip a fair coin 10 times and count the number of times it lands on tails. 50 people are asked to conduct the experiment and the mean number of tails each person obtained is calculated, ̅ X.
(iii)
The lengths of time taken by university students to complete a crossword puzzle follows a normal distribution. A random sample of 80 students is taken and the mean of their times taken to complete the crossword puzzle is calculated, ̅ X .
(iv)
A fair die is rolled, and George records the number of rolls taken before he obtains a 6. George repeats this process 40 times and calculates the mean number of rolls taken before obtaining a 6.

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6a
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The standard normal distribution is given by Z space tilde space N left parenthesis 0 comma 1 squared right parenthesis.

Given that P space left parenthesis negative a less than Z less than a right parenthesis equals 0.95,

(i)
explain why P space left parenthesis Z less than a right parenthesis equals 0.975,
(ii)
hence find the value of a.

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6b
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A random sample of 25 observations is taken from N space left parenthesis mu comma 6 ² right parenthesis and the sample mean, ̅ x is 15. The formula for the end-points of the 95% confidence interval for the population mean, mu is given by:

̅ x space plus-or-minus space z cross times fraction numerator sigma over denominator square root of n end fraction space space space w h e r e space P space left parenthesis negative z less than Z less than z right parenthesis equals 0.95

Using your answer to part (a) and the formula above

(i)
Calculate the end-points of the 95% confidence interval for mu,
(ii)
Hence write down the 95% confidence interval for mu.

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6c
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Write down the probability that the confidence interval found in part (b) contains the true population mean, mu.

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7a
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The standard normal distribution is given by Z space tilde space N left parenthesis 0 comma 1 squared right parenthesis.

Given that P space left parenthesis negative a less than Z less than a right parenthesis equals 0.99,

(i)
Explain why P space left parenthesis Z less than a right parenthesis equals 0.995,
(ii)
Hence find the value of a.

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7b
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The probability of success in a trial is p. A random sample of 64 trials are carried out and there were 24 successes.

Calculate the proportion of the trials in the sample, p subscript s, that were successful.

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7c
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The formula for the end-points of the 99% confidence interval for the probability of success, p is given by:

p subscript s plus-or-minus z cross times square root of fraction numerator p subscript s left parenthesis 1 minus p subscript s right parenthesis over denominator n end fraction end root space space space space space space space w h e r e space P space left parenthesis negative z less than Z less than z right parenthesis equals 0.99

Calculate an approximate 99% confidence interval for the probability of success in a trial, p.

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1a
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The times, T seconds, of a random sample of 8 TV adverts on Channel π are as follows.

35

61

42

40

37

35

53

49

Calculate unbiased estimates of the population mean and variance of T.

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1b
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It is known that the population of T follows a Normal distribution with a variance of 81.

Using the sample from part (a), find a 95% confidence interval for the population mean of T.

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1c
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Write down the probability that the confidence interval contains the true population mean.

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2a
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The masses, M grams, of a random sample of 80 potatoes from a farm are summarised as follows.

n equals 80 space space space space space space space space space space space space sum m equals 17764 space space space space space space space space space space space sum m squared equals 4103225

Calculate unbiased estimates of the population mean and variance.

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2b
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Calculate a 90% confidence interval for the population mean.

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2c
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Explain why it was necessary to use the Central Limit theorem in your answer to part (b).

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3a
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The gestation period of a female kangaroo, X can be modelled as a Normal distribution with a mean of 29 days and a standard deviation of 4 days.

Given that a randomly selected female kangaroo is pregnant, find the probability that the gestation period will be between 27 and 30 days.

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3b
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A random sample of 16 pregnant kangaroos is taken and the mean of their gestation periods is calculated.

(i)
Write down the distribution of the sample mean, ̅ X.
(ii)
Calculate the probability that the sample mean of the 16 gestation periods is between 27 and 30 days.

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3c
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Explain whether it was necessary to use the Central Limit theorem in the solution to part (b).

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4a
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For the video game, Super Maria, it is known that the length of time, T minutes, it takes a gamer to complete the final level of the game can be modelled as a Normal distribution with T space tilde space N left parenthesis 57.2 comma 5 squared right parenthesis.

Find the interquartile range for the times taken to complete the final level of Super Maria.

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4b
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During a Super Maria competition, gamers are randomly put into teams of 9 and each member plays the final level. The mean time for each team is calculated and prizes are given to teams whose means are in the fastest 10% of mean times.

(i)
Write down the distribution of the sample mean, ̅ T.
(ii)
Find, to the nearest second, the maximum mean time that would lead to a team winning a prize. Give your answer in minutes and seconds.

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5a
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A spinner has 5 sectors labelled 1 to 5. The spinner is spun 150 times and there were 131 occasions when it landed on the number 3.

Calculate an approximate 92% confidence interval for p, the probability that the spinner lands on the number 3.

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5b
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Calculate the width of the confidence interval to 3 significant figures.

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5c
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Explain why the method in part (a) would not be suitable if the spinner had only been spun 15 times.

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6a
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ChocoCakes sells chocolate brownies and claims they are good value for money.  Sasha, an inspector, takes a random sample of 40 brownies and records the mass, M grams, of each one.  The results are summarised as follows.