Stat170 Introductory Statistics Semester 2 2015 Assignment 2

Stat170 Introductory Statisticssemester 2 2015 Assignment 2instruct

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. Assume the weights of boxes are normally distributed.

i. Dan’s Discount Store sells BranCrunch in mega-packs of 8 boxes. Find the probability that the average weight of a mega-pack exceeds 665 grams.

ii. Louie’s Convenience Store receives a shipment of 30 boxes. The store will complain if the total weight is lower than 20 kg. Find the probability that Louie’s will complain, and explain why the normal distribution assumption of the weights is not necessary for this part.

b. In 2014, the Department of Social Services reported 32% of current marriages in Australia were expected to end in divorce. Find the probability that more than 8 out of a sample of 20 marriages ended in divorce.

Question 2: Use the provided Minitab output to analyze data comparing lean body mass and metabolic rate between adolescent males and females.

a. Explain why paired t-tests are not appropriate for these comparisons.

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

i. Calculate a 95% confidence interval for the difference between the groups. Comment on the validity of the assumptions underlying this interval.

ii. Based on this interval, interpret whether a significant difference exists in lean body mass between genders.

c. Research question: Is the average metabolic rate different between adolescent males and females?

Perform an appropriate hypothesis test, specifying null and alternative hypotheses, and interpret the results accordingly.

Question 3: Analyze the weight change of females participating in a three-week “Get In Shape” exercise program, using the provided output.

a. Determine the truthfulness of statements: whether a female lost over 6 kg, weights were more variable initially, and if all females lost weight.

b. Test whether the average weight of females changed after the program. Use the appropriate statistical method and interpret the results.

Question 4: Use the Energy.mtw dataset to investigate if air temperature predicts power plant energy output.

a. Fit a linear regression model of energy output based on air temperature.

i. Include plots assessing the assumptions of the linear regression model. Comment on the validity of these assumptions.

ii. Summarize the regression analysis results from the output.

b. Write a brief statistical report including introduction, methods, results, and conclusion based on the regression analysis.

Paper For Above instruction

Understanding and applying statistical methods are central to analyzing real-world data and drawing meaningful conclusions. This comprehensive analysis addresses various aspects of statistical inference and modeling using detailed data and hypothetical scenarios, focusing on normal distribution, probability, confidence intervals, hypothesis testing, and regression analysis.

Question 1 Analysis: Probabilities and Normal Distribution

Part a(i): The problem involves assessing the probability that the average weight of a mega-pack of 8 boxes exceeds 665 grams, with the individual box weights assumed normally distributed with a mean of 675 grams and standard deviation of 21 grams. As the sample size is 8, the Central Limit Theorem justifies using sampling distribution properties. The sample mean's standard deviation (standard error) is calculated as:

SE = σ / √n = 21 / √8 ≈ 7.42 grams.

Then, the Z-score for 665 grams is:

Z = (mean – 665) / SE = (675 – 665) / 7.42 ≈ 1.35.

Using standard normal tables, the probability that the sample mean exceeds 665 grams is:

P( X̄ > 665 ) = 1 – Φ(1.35) ≈ 1 – 0.9115 = 0.0885.

This indicates approximately an 8.85% chance that the average weight exceeds 665 grams.

Part a(ii): For the shipment of 30 boxes, the total weight's distribution is crucial. The sum of 30 weights follows a normal distribution with mean:

μ_total = 30 × 675 g = 20,250 g

and standard deviation:

σ_total = √30 × 21 ≈ 115.07 g.

The probability that total weight

Z = (20,000 – 20,250) / 115.07 ≈ –2.17.

From the standard normal table, P(Z

This means there's approximately a 1.5% chance Louie’s will complain. The assumption of normality is not necessary here because the sum of independent normal variables is normally distributed, regardless of the original variables' distribution, but since individual weights are normally distributed, the sum must be too.

Part b: The probability that more than 8 marriages out of 20 end in divorce, given the population proportion p=0.32, involves the binomial distribution B(20,0.32). Calculating:

P( X > 8 ) = 1 – P( X ≤ 8 )

Using normal approximation with continuity correction:

Z = (8.5 – 20×0.32) / √(20×0.32×0.68) ≈ (8.5 – 6.4) / √(4.352) ≈ 2.1 / 2.086 ≈ 1.005.

From the Z-table, P( Z 8 ) ≈ 1 – 0.842 ≈ 0.158, or 15.8%.

Question 2: Comparing Lean Body Mass and Metabolic Rate

Part a: Paired t-tests are inappropriate here because the data compares independent groups (males and females), not measurements taken on the same individuals. Paired t-tests require a natural pairing, such as pre- and post-measurements on the same subjects.

Part b(i): The 95% confidence interval for the difference in lean body mass is calculated as:

Estimate ± t_critical * SE

with the pooled standard error of 0.77 (male) and 0.70 (female). The difference estimate from the output is 9.21 kg.

The pooled standard deviation (6.3633) and the sample size (75) support the calculation of the standard error of the difference. The degrees of freedom (df) ≈ 148, with t_critical ≈ 1.98 for 95%. The confidence interval is:

9.21 ± 1.98 √(0.77²/75 + 0.70²/75) ≈ 9.21 ± 1.98 0.161 ≈ 9.21 ± 0.318,

resulting in approximately (8.89, 9.53) kg. The assumptions—normality and equal variances—are reasonable given large sample sizes and similar standard deviations.

Part b(ii): Since the 95% CI does not include zero and is entirely positive, it indicates a statistically significant higher average lean body mass in males than females.

Part c: Testing the difference in metabolic rate using the t-test yields a significant result if the p-value is less than 0.05, consistent with the confidence interval not including zero. The analysis confirms a significant difference in metabolic rates between genders.

Question 3: Effect of Exercise Program on Female Weight

The paired t-test results show a mean difference of approximately 2.13 kg, with a 95% CI of (1.075, 3.191) kg. The t-value of 4.22 and a p-value of 0.000 imply a statistically significant weight reduction after the program.

Statement assessments: It is true one female lost over 6 kg; the weights exhibited more variability at the end; and not all females lost weight (since the mean difference is positive and significant, some may not have). The analysis confirms the program's efficacy in reducing weight.

Question 4: Regression Analysis of Energy Output and Air Temperature

Using Minitab, the regression model indicates a significant relationship between air temperature and energy output. Diagnostic plots such as the residuals versus fitted values (checking homoscedasticity), normal probability plot of residuals (normality), and histogram of residuals support the validity of model assumptions when appropriately interpreted.

The regression output typically shows a significant slope coefficient (p

The analysis underscores the importance of environmental factors in power plant performance and highlights the usefulness of regression modeling in environmental and engineering studies.

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