L3.2 Confidence Intervals Starters

Starters A B C D E F G H I J K L M N

Starter A Home

A population of little pigs have a average weight of 24.3kg (standard deviation of 4.2kg)

A wolf caught a random sample of 3 pigs

1) What is the likely total weight of the three pigs?

2) What is the standard deviation of the total weight of the 3 pigs?

3) What is the probability that the total weight is over 65kg?

Starter A Answers Home

A population of little pigs have a average weight of 24.3kg (standard deviation of 4.2kg)

A wolf caught a random sample of 3 pigs

1) What is the likely total weight of the three pigs?

2) What is the standard deviation of the total weight of the 3 pigs?

3) What is the probability that the total weight is over 65kg?

Starter B Home

A population of little pigs have a average weight of 26.8kg (standard deviation of 3.5kg) (they have put on weight since yesterday)

A wolf once again caught a random sample of 3 pigs

1) What is the likely average weight of the three pigs?

2) What is the standard deviation of the average weight of the 3 pigs?

3) What is the probability that the average weight is under25kg?

Starter B Answers Home

A population of little pigs have a average weight of 26.8kg (standard deviation of 3.5kg) (they have put on weight since yesterday)

A wolf once again caught a random sample of 3 pigs

1) What is the likely average weight of the three pigs?

2) What is the standard deviation of the average weight of the 3 pigs?

3) What is the probability that the average weight is under25kg?

Starter C Home

A population of little pigs have a average weight of 29.4kg (standard deviation of 3.2kg) (more weight gain)

A wolf once again caught a random sample of 3 pigs and he put them in a box weighing 8.5kg to freight them home.

1) What is the average weight of the box containing the pigs?

2) What is the standard deviation of the box containing the pigs?

3) What is the probability that the box containing the pigs at most 100kg?

Starter C Answers Home

A population of little pigs have a average weight of 29.4kg (standard deviation of 3.2kg) (more weight gain)

A wolf once again caught a random sample of 3 pigs and he put them in a box weighing 8.5kg to freight them home.

1) What is the average weight of the box containing the pigs?

2) What is the standard deviation of the box containing the pigs?

3) What is the probability that the box containing the pigs at most 100kg?

Starter D Home

One little pig (who is about to be converted into a selection of pork products) decides to build a house consisting of 20 sticks. The average weight of a stick is 1.8kg (standard deviation of 0.23kg)

The sticks were measured to the nearest 0.1kg

1) What is the average total weight of the stick house?

2) What is the variance of the total weight of the stick house?

3) What is the probability that total weight of the stick house is under 35 kg?

Starter D Answers Home

One little pig (who is about to be converted into a selection of pork products) decides to build a house consisting of 20 sticks. The average weight of a stick is 1.8kg (standard deviation of 0.23kg)

The sticks were measured to the nearest 0.1kg

1) What is the average total weight of the stick house?

2) What is the variance of the total weight of the stick house?

3) What is the probability that total weight of the stick house is under 35 kg?

Starter E Home

As the wolf was eating the little pig from the stick house he noted that each mouthful of flesh was independent had an average weight of 240g. He also knew that the standard deviation of all his previous mouthfuls was 45g and he had eaten 34 mouthfuls of pig for dinner. He was clever.

1) If the wolf was to be 95% sure of the average mouthful weight of all previous mouthfuls, what would the upper and lower weight limits be?

2) What is the margin of error for the 95% confidence interval for wolf mouthful size?

Starter E Answers Home

As the wolf was eating the little pig from the stick house he noted that each mouthful of flesh was independent had an average weight of 240g. He also knew that the standard deviation of all his previous mouthfuls was 45g and he had eaten 34 mouthfuls of pig for dinner. He was clever.

1) If the wolf was to be 95% sure of the average mouthful weight of all previous mouthfuls, what would the upper and lower weight limits be?

2) What is the margin of error for the 95% confidence interval for wolf mouthful size?

Starter F Home

The wolf was tired of pigs so he went to steal Little-Red-Hooding-Rides basket of scones again. He knew the standard deviation of all the scones was 64g but he had forgotten the average scone weight. After terrorizing Hooding-Ride once again he measure her basket of 12 scones and found the average weight was 225g

1) What is the 99% confidence interval for the mean weight of all scones?

2) Little-Red-Hooding-Ride had said the average scone weight was 275g. Is she a lying little...#*!? Give numerical evidence

Starter F Answers Home

The wolf was tired of pigs so he went to steal Little-Red-Hooding-Rides basket of scones again. He knew the standard deviation of all the scones was 64g but he had forgotten the average scone weight. After terrorizing Hooding-Ride once again he measure her basket of 12 scones and found the average weight was 225g

1) What is the 99% confidence interval for the mean weight of all scones?

2) Little-Red-Hooding-Ride had said the average scone weight was 275g. Is she a lying little...#*!? Give numerical evidence

Starter G Home

The wolf wanted to compare his impressive teeth with those of other wolves. The wolverine magazine described all wolf teeth having a standard deviation of 5mm in length.

1) How many of his own teeth will the wolf need to extract if the mean is to be estimated within 2mm with a 95% level of confidence.

2) What is the margin of error for the confidence interval for tooth length?

Starter G Answers Home

The wolf wanted to compare his impressive teeth with those of other wolves. The wolverine magazine described all wolf teeth having a standard deviation of 5mm in length.

1) How many of his own teeth will the wolf need to extract if the mean is to be estimated within 2mm with a 95% level of confidence.

2) What is the margin of error for the confidence interval for tooth length?

Starter H Home

The wolf made a comparison between straw and stick housed pigs:

He ate 14 straw house pigs and found a mean weight of 26.5kg (Population Standard deviation = 2.8kg)

He ate 10 stick house pigs and found a mean weight of 28.4kg (Population Standard deviation = 2.2kg)

1) Calculate the 95% confidence interval for the difference between the population means.

2) Are the straw house and stick house pigs significantly different at the 95% level of confidence. Why?

Starter H Answers Home

The wolf made a comparison between straw and stick housed pigs:

He ate 14 straw house pigs and found a mean weight of 26.5kg (Population Standard deviation = 2.8kg)

He ate 10 stick house pigs and found a mean weight of 28.4kg (Population Standard deviation = 2.2kg)

1) Calculate the 95% confidence interval for the difference between the population means.

2) Are the straw house and stick house pigs significantly different at the 95% level of confidence. Why?

Starter I Home

The wolf caught 30 little pigs and found that 12 of them had swine flu and were not fit for consumption as they were all snotty.

1) Calculate the 90% confidence interval for proportion of pigs with swine flu.

2) What is the margin of error for the 95% confidence interval for proportion of pigs with swine flu.

Starter I Answers Home

The wolf caught 30 little pigs and found that 12 of them had swine flu and were not fit for consumption as they were all snotty.

1) Calculate the 90% confidence interval for proportion of pigs with swine flu.

2) What is the margin of error for the 95% confidence interval for proportion of pigs with swine flu.

Starter J Home

Last year there were 32% of pigs has swine flu

1) How many pigs does the wolf need to sample to get a estimate of the percentage of pigs with swine flu this year with an accuracy of 4%? (at the 90% level of confidence)

2) What is the maximum margin of error for of the proportion of pigs with swine flu this year if 200 were sampled? (at the 90% level of confidence)

Starter J Answers Home

Last year there were 32% of pigs has swine flu

1) How many pigs does the wolf need to sample to get a estimate of the percentage of pigs with swine flu this year with an accuracy of 4%? (at the 90% level of confidence)

2) What is the maximum margin of error for of the proportion of pigs with swine flu this year if 200 were sampled? (at the 90% level of confidence)

Starter K Home

1) In the sample of 90 trees the nursery manager discovers that leaf-spot, a disease caused by a fungus, is present in 10% of them. Assuming that the leaf-spot fungus occurs randomly in trees, find a 99% confidence interval for the proportion of all trees currently in stock that have leaf-spot.

2) The manager requires a more accurate indication of the proportion of all trees in stock that have leaf-spot, and decides to take another sample.

She wants to know the proportion of all trees in stock with leaf-spot, estimated to within 4% of the true value with a 99% level of confidence.

Find the minimum sample size that is required to meet this condition.

Starter K Answers Home

1) In the sample of 90 trees the nursery manager discovers that leaf-spot, a disease caused by a fungus, is present in 10% of them. Assuming that the leaf-spot fungus occurs randomly in trees, find a 99% confidence interval for the proportion of all trees currently in stock that have leaf-spot.

2) She wants to know the proportion of all trees in stock with leaf-spot, estimated to within 4% of the true value with a 99% level of confidence.

Find the minimum sample size that is required to meet this condition.

2.576 0.5 0.5

n0.04

n 1036.84

2.576 0.10.9

n0.04

n 373.26Minimum = 1037 OR

Minimum = 374

0.1 0.081 OR0.0185 0.181

Starter L Home

Let μ1 be the mean height of trees in the 2008 stock analysis sample and let μ2 be the mean height of trees in the 2005 stock analysis sample.

1) Find a 95% confidence interval for μ1– μ2

2) Explain, in terms of the confidence interval in part (a), whether the manager’s belief is justified that the new potting mix has made the trees grow taller.

2008 tree analysis

2005 tree analysis

Sample size (number of trees) 90 75

Sample mean height (cm) 142.3 138.4

Sample standard deviation (cm) 13.2 10.9

Starter L Answers Home

Let μ1 be the mean height of trees in the 2008 stock analysis sample and let μ2 be the mean height of trees in the 2005 stock analysis sample.

1) Find a 95% confidence interval for μ1– μ2

2) Explain, in terms of the confidence interval in part (a), whether the manager’s belief is justified that the new potting mix has made the trees grow taller.

2008 tree analysis

2005 tree analysis

Sample size (number of trees) 90 75

Sample mean height (cm) 142.3 138.4

Sample standard deviation (cm) 13.2 10.9

The manager’s belief is justified because: zero lies outside this interval OR the whole interval is positive.

3.9 3.68 OR 0.22 μ1– μ2 7.58

7.58

Starter M Home

The girth of each of the 90 trees in a sample is measured and the following 96% confidence interval is obtained for μ, the mean girth of all trees in stock: 8.3 cm ≤ μ ≤ 34.1 cm

1) Explain, in terms of the girth of all trees in stock, the meaning of this confidence interval.

2) If the sample size had been k times greater, but the sample mean, sample standard deviation and confidence level had stayed the same, what would have been the width of the interval obtained?

Starter M Answers Home

1) Explain, in terms of the girth of all trees in stock, the meaning of this confidence interval.

2) If the sample size had been k times greater, but the sample mean, sample standard deviation and confidence level had stayed the same, what would have been the width of the interval obtained?

k

125.8

kCurrent interval width = 25.8 cm.New interval width is

times the original interval width.So new interval width is

.

There is a 96% chance that the interval contains the mean girth of all trees in the nursery. ORIf the sampling process was repeated a large number of times, 96% of such intervals would contain the mean girth of all trees in the nursery.NOT!!!“There is a 96% chance that the population mean is within this interval”

Starter N (excellence) Home

1) The nursery manager predicts that the mean height of all the trees in stock will be 150 cm.If this prediction is correct, calculate the probability that a random sample of 90 trees from the nursery’s stock would produce a mean height of 142.3 cm or less, and comment on what that would imply about such a sample.(Note: use the sample standard deviation of 13.2 cm as an estimate of the population standard deviation.)

2) The mean weight of the potted trees in the sample is 38.7 kg, with a standard deviation of 2.4 kg.A trailer is to be loaded with 12 of these trees. Calculate a 95% confidence interval for T, the true total weight of a trailer-load of 12 trees.

Starter N (excellence) Answers Home

1) The nursery manager predicts that the mean height of all the trees in stock will be 150 cm.If this prediction is correct, calculate the probability that a random sample of 90 trees from the nursery’s stock would produce a mean height of 142.3 cm or less, and comment on what that would imply about such a sample.(Note: use the sample standard deviation of 13.2 cm as an estimate of the population standard deviation.)

2) The mean weight of the potted trees in the sample is 38.7 kg, with a standard deviation of 2.4 kg.A trailer is to be loaded with 12 of these trees. Calculate a 95% confidence interval for T, the true total weight of a trailer-load of 12 trees.

E[T ] 12 38.7 464.4

T 12 2.4 8.3138

Confidence Interval:464.4 16.295 OR

448.1 T 480.7

391402.190

2.13X

0)3.142(P X

It is highly unlikely that a sample would be obtained with a mean height of 142.3 cm or less.