Central Limit Theorem
Questions 1,2,3 a & c, 8
File #1
1. A random sample of 20 students had a mean height of 1.74 metres These heights are normally distributed
with a standard deviation of 0.1 metres
a) Find a symmetrical 95% confidence interval for the mean height of students
[5]
b) Explain what is meant by a 95% confidence interval for the mean.
[1]
2. a) Explain what is meant by the Central Limit Theorem
[2]
The weights of a species of bird have mean 16g and standard deviation 0.8g.
b) State the distribution of the mean weight of a sample of 50 such birds
[2]
c) Find the probability that the mean weight of such a sample exceeds 16.2g
[4]
3. An article suggests that students are substantially more likely to be overweight than the rest of the population.
To test this, a sample of N students is chosen at random and their weights recorded.
Assuming that the standard deviation of the weights of the student population is 9.7kg,
a) Find, in terms of N, the width of a 95% confidence interval for the mean weight of the students.
[4]
It is decided that the width of the confidence interval should be at most 0.5kg.
b) Find the minimum possible value of N
[3]
Assuming N takes this minimum value, and that the sample mean was 60.4kg,
c) Write down the 95% confidence interval for the mean weight of the students
[2]
In the population as a whole, the mean weight is 60.1kg
d) State and explain any conclusions that can be drawn from the results on the sample of students.
[2]
4. The mean mark in one module of an examination is 63, with a variance of 36.
50 examination centres each enter 120 students for this module.
a) Find the expected number of centres for which the mean mark exceeds 64.
[7]
b) State two assumptions you have used in your calculation.
[2]
5. The length of a particular type of nail is known to be normally distributed with mean ? cm.
A sample of nails is taken in order to determine a confidence interval for ?.
Given that the sample mean was 3.05cm and the standard error of the mean is 0.01,
a) Find a 95% confidence interval for ?.
[3]
An ?% confidence interval is calculated. It has width 0.047cm.
b) Find ?, giving your answer to the nearest whole number
[6]
The manufacturer claims that ? = 3.04cm.
c) Explain whether the sample provides sufficient evidence to reject this claim.
[2]
6. Scores in an IQ test have been found to be normally distributed with mean 100 and standard deviation 22.9.
This test is given to a random sample of 200 people.
a) Find the probability that their mean score is within 1 of the population mean
[6]
b) Find the score that the sample mean will exceed with a probability of 0.01
[3]
In fact, the sample mean was found to be 106.
c) Comment on this result
[2]
7. A factory produces widgets, of which 15% are faulty. They are packed in boxes of forty.
a) State the mean and variance of the number of faulty widgets in a box.
[3]
A sample of 200 boxes is taken.
b) Find the probability that the mean number of faulty widgets per box is between 5.5 and 6.5
[6]
c) State one assumption required for your calculation in b)
[1]
8. The diameters of cakes at a bakery are normally distributed with mean ?cm and variance 0.81cm2.
A sample of 50 cakes is found to have mean diameter 17.8cm.
a) Find a symmetrical 99% confidence interval for ?
[4]
100 samples of 50 cakes were taken.
b) Find how many of these samples would be expected to produce a 99% confidence interval for the mean
which did not contain ?.
[2]
c) Find the probability that all 100 samples produced confidence intervals containing ?.
[2]
9. An unbiased, tetrahedral dice has the numbers 1, 2, 3, 4 on its four faces.
a) Find the mean and variance of the score obtained when the dice is thrown
[4]
The dice is thrown 200 times
b) Making a suitable assumption, which you should state, write down the distribution of the mean score
obtained from the 200 throws.
[4]
c) Find a symmetric 95% confidence interval for the mean score of the 200 throws.
[4]
10. A researcher obtained details on the salaries of 100 married couples. She then noted the figure obtained by subtracting the earnings of the wife from the earnings of the husband.
In the sample, the mean and standard deviation of these figures were £63 and £248.75 respectively.
a) Find an unbiased estimate for the population standard deviation
[3]
b) Find a 99% confidence interval for the difference between husbands’ and wives’ salaries
[4]
c) Hence comment on the assertion that husbands earn significantly more than wives, explaining the reasons
for your comments.
[2]
11. Tesbury’s supermarket claim that their boxes of strawberries are better value than those of their rivals Sainsco.
To test this, a researcher obtained a sample of 50 boxes from Tesbury’s and 60 boxes from Sainsco, and recorded their weights (in grammes) . The results are shown in the table below:
Tesbury’s Sainsco
?x 12550 15055
?x2 3,150,491 3,777,919
a) Find estimates for the variance of the weights of the boxes sold at each of the supermarkets
[4]
b) Find a 95% confidence interval for the difference of the weights of the boxes
[5]
c) Hence comment on Tesbury’s claim, giving a reason for your answer.
[2]
d) State one assumption required for your calculations
[1]
12. For advertising purposes, “Kickin’ Kola” decide to find an estimate for the proportion of the population
who prefer their drink to their nearest rival, “Peppy Kola”. They decide to take a sample of size 10,000.
Assuming the proportion of the population who prefer “Kickin’ Kola” is p,
a) State the mean and variance of the number of people in the sample who prefer “Kickin’ Kola” .
[3]
b) Hence or otherwise, write down the mean and variance of the proportion of the sample who prefer
“Kickin’ Kola”
[2]
Given that 5320 of the sample preferred “Kickin’ Kola”,
c) Find a 95% confidence interval for p.
[5]
“Peppy Kola” claim that 55% of the population prefer their product.
d) Use your answer to c) to comment on this assertion.
[2]
13. The heights of eight year olds are normally distributed with mean ? and variance ?2.
a) Write down the distribution of , the mean height of a sample of N eight year olds.
[2]
This sample is to be used to obtain a 95% confidence interval for ?. This interval is required to be at most
0.01? in width.
b) Find the minimum possible value of N
[4]
A second sample records the heights of eight year old girls and eight year old boys separately
The standard error of the difference between the means of the boys and the girls is 0.02cm, and
the mean heights of the sample of boys and the sample of girls respectively were 1.079m and 1.074m.
c) Find a 99% confidence interval for the difference in mean heights of boys and girls.
[4]
d) Explain the significance of whether or not zero is included in your confidence interval.
[1]
14. The number of holes per 10m2 of a particular brand of carpet follows a Poisson distribution with mean ?.
One hundred hotel rooms, each of area 10m2, were carpeted with this brand.
a) State the variance of the number of holes per room
[1]
Assuming that the sample is sufficiently large for the Central Limit Theorem to apply,
b) State the distribution of the mean number of holes per room
[2]
Given that there were an average of 16 holes per room,
c) Find a 98% confidence interval for ?.
[4]
6.
E-mail assignment
Assignment
For this assignment, you need to create a response to each of the email messages provided.
The purpose of this assignment is for you to apply writing principles while responding to business email. The emails you’ll receive simulate internal and external. Assume that you are employed at the Blue Bay Resort and Conference Center. As you might expect, the culture is fast-paced and fairly informal, with employees relying heavily on email. While speed is important, high value is also placed on precision: All of your email responses should be grammatically correct and error free.
You will play the role of the catering director for the Blue Bay Resort and Conference Center. You took this job five years ago and are busier than you ever imagined you might be—running from meeting to meeting every day. Your only time to respond to your email is during your short breaks between customer meetings. As usual, you have a flood of messages which all seem critical. Your challenge during this assignment is to respond to all of your email and write clear responses.
When responding to each of the messages, consider all of the following:
What do you think is an appropriate response to your message?
What information do you think should be included?
What might be inappropriate for this type of message?
How long should an appropriate response be?
What would you consider too long?
What tone should you use for the message?
