STA201 - Statistical Inference - Employ Sampling Distributions to Find Probabilitie - Statistics Assignment Help

Assignment Task

Introduction
In this assignment you need to use the techniques of Topics 4 - 7 to employ sampling distributions to find probabilities for means and proportions, and to make inferences about one and two samples.
We estimate that this assignment will take you about six hours to do once you are properly prepared for it.

Rationale
Assignment 2 is designed to assess the following learning outcomes:
• Able to use a statistical package to analyse data appropriately, and then interpret the output.
• Able to calculate and interpret probabilities, and use standard discrete and continuous probability distributions
• Able to explain the concepts of statistical inference, and apply these to confidence intervals and tests of hypotheses.
• Able to evaluate if the assumptions underlying statistical techniques are valid in a given scenario.

Throughout the assignment you are expected to use R Commander to calculate probabilities, analyse data, perform statistical inference and produce plots unless you are directed to use manual calculations.

Question 1 (7 marks)

Let X be the live weight of a randomly selected 4-month old male lamb. As in Question 5 inAssignment 1, suppose X follows a normal distribution with a mean of 38 kg and a standard deviation of 2.5 kg. In this question, we consider a sample of sixteen 4-month old male lambs. Denote by X ? the mean live weight of these 16 lambs.

(a) State the distribution, with the corresponding parameters, of X ?. [2 marks]
(b) Manually calculate the probability that the mean live weight of these 16 lambs is greater than 39 kg. [3 marks]
(c) An important assumption of the Central Limit Theorem [which is used to derive the distribution of X ? in (a)] is independence. In the context of this question, suggest a situation where the independence assumption may be violated.

Question 2 (10 marks)
This question is testing your understanding of some important concepts about hypothesis testing and confidence intervals. For each part below, you must explain your answer.

(a) Suppose we are performing a one-sample t test at the 10% level of significance where the hypotheses are H0 : μ = 0 vs H1 : μ 6= 0. The number of observations is 15. What is the critical value? [2 marks]

(b) Suppose we are performing a one-sample t test with H0 : μ = 0 vs H1 : μ > 0. The test statistic, 1.31, was found to be wrongly calculated. The correct test statistic should be 2.31 instead. Is the correct p-value higher than, lower than, or the same as the original one? [2 marks]

(c) Suppose we are performing a hypothesis test and we conclude that we can reject H0 at the 10% level of significance, can we reject the same H0 (with the same H1) at the 1% level of significance?

[2 marks]
(d) Determine if the following statement is true or false: [2 marks] H0 can always be rejected if the observed test statistic is greater than the critical value.
(e) Based on the data, we obtain (0.2, 0.6) as the 95% confidence interval for the true mean. What can we say about the test with the hypotheses H0 : μ = 0 against H1 : μ 6= 0 at the 5% level of significance?

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