Statistics

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1)

Question 1 (10 marks)

Include an appropriate diagram for each part of this question. You can sketch these diagrams and paste in a photo of your sketch, along with your solutions.

a. BranCrunch is a new breakfast cereal. Boxes of BranCrunch are labelled ‘ 675 grams’ but there is some variation. The actual mean weight is 675 grams with a standard deviation of 21 grams.

i. Dan’s Discount Store sells BranCrunch in mega-packs of 8 boxes. Assuming the weights of boxes of BranCrunch are normally distributed, find the probability that the average weight of a mega-pack of BranCrunch is higher than 665 grams.

ii. Louie’s Convenience Store receives a shipment of 30 boxes of BranCrunch. Louie’s will complain to the manufacturers if the total weight of this shipment is lower than 20 kg. Find the probability that Louie’s Convenience Store will complain about the shipment and explain why the information about the weights of BranCrunch following a normal distribution is not necessary to answer this part of the question.

b. In 2014 the Department of Social Services reported that 32% of current marriages in Australia were expected to end in divorce.

Find the probability that more than 8 marriages out of a random sample of 20 marriages which were current in 2014 would end in divorce.

Question 2 (13 marks)

Lean body mass is the amount of weight carried on the body that is not fat. Metabolic rate is the rate at which the body consumes energy. The following Minitab output was constructed using data recorded in a fitness study which was designed to compare the lean body masses and also the metabolic rates of adolescent males and adolescent females. Use this output to answer the questions which follow.

Two-sample T for LeanBodyMass

Sex N Mean StDev SE Mean

Male 75 52.63 6.66 0.77

Female 75 43.42 6.05 0.70

Difference = μ (Male) - μ (Female)

Estimate for difference: 9.21

95% CI for difference: (****, ****)

T-Test of difference = 0 (vs ≠): T-Value = **** P-Value = **** DF = ****

Both use Pooled StDev = 6.3633

Two-sample T for MetRate

SE

Sex N Mean StDev Mean

Male 75 1626 227 26

Female 75 1258 172 20

Difference = μ (Male) - μ (Female)

Estimate for difference: 367.9

95% CI for difference: (****, ****)

T-Test of difference = 0 (vs ≠): T-Value = **** P-Value = **** DF = ****

Both use Pooled StDev = 201.5088

a. Explain why you could not use paired t-tests for these comparisons: Question 2 continued

b.

Research Question: Is there a difference between the average lean body mass of adolescent males and adolescent females?

i. Calculate a 95% confidence interval to estimate the difference between the average lean body mass of adolescent males and the lean body mass of adolescent females. You should comment on the validity of any underlying assumptions.

ii. Use this interval to comment on the research question above.

c.

Research Question: Is there a difference between the average metabolic rate of adolescent males and adolescent females?

c.

Metabolic rate is the rate at which the body consumes energy. It is an important variable in studies of weight gain, exercise and health. Use an appropriate hypothesis test to answer this research question.

Hypothesis Test:

Question 3 (10 marks)

20 females took part in a three week exercise programme called “Get In Shape”. As part of an evaluation of the “Get In Shape” programme, weights for each of these 20 females were measured both at the beginning and at the end of the programme. The output below was produced from the results of this evaluation. Use the most appropriate part/s of this output to answer the questions which follow.

Two-Sample T-Test and CI: WeightBeginning, WeightEnd

Two-sample T for WeightBeginning vs WeightEnd

N Mean StDev SE Mean

WeightBeginning 20 64.74 5.99 1.3

WeightEnd 20 62.61 6.56 1.5

Difference = μ (WeightBeginning) - μ (WeightEnd)

Estimate for difference: 2.13

95% CI for difference: (-1.89, 6.15)

T-Test of difference = 0 (vs ≠): T-Value = 1.07 P-Value = 0.290 DF = 38

Both use Pooled StDev = 6.2809

Paired T-Test and CI: WeightBeginning, WeightEnd

Paired T for WeightBeginning - WeightEnd

N Mean StDev SE Mean

WeightBeginning 20 64.74 5.99 1.34

WeightEnd 20 62.61 6.56 1.47

Difference 20 2.133 2.260 0.505

95% CI for mean difference: (1.075, 3.191)

T-Test of mean difference = 0 (vs ≠ 0): T-Value = 4.22 P-Value = 0.000

Question 3 continued

a.

b.

Research Question: On average, do adolescent females who take part in the “Get in Shape” programme experience a significant change in their body weights?

Using the appropriate part/s of the output on the previous page, carry out a hypothesis test to answer this research question.

Hypothesis Test:

Question 4 (17 marks)

The Energy study was conducted over a six year period to investigate factors which could be used to predict the net hourly energy output of a power plant. The Minitab file, Energy.mtw, which is on iLearn consists of a randomly selected sample of 100 observations recorded during the course of the study. Each observation consists of information recorded on the following variables.

Variable Description

AirTemp Hourly average air temperature (°C)

Energy Net hourly electrical energy output (megawatts)

Research Question: Is air temperature a useful predictor of energy output?

a. Use the Minitab file, Energy.mtw, to fit a linear regression model for predicting energy output from air temperature. Cut and paste the following output into the space below:

i. Appropriate Minitab plots for checking each of the three assumptions of a linear model. Do not use the ‘four in one’ graph option in Minitab. Comments on each of these assumptions should go into your report on the following page.

ii. The regression analysis of Energy vs AirTemp from Minitab

Question 4 continued

b. Write a short statistical report on the regression analysis from part a.

Introduction:

Methods:

Results:

Conclusion:

Assignment 1

1

STAT170 Assignment 2 Session 2 2015

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Histogram of MetRate

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Histogram of WeightBeginning, WeightEnd, Weight Differences

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Boxplot of LeanBodyMass