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Statistical Sciences 2141A/B Lecture Notes - Lecture 25: Central Limit Theorem, Confidence Interval

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School
Western University
Department
Statistical Sciences
Course
Statistical Sciences 2141A/B
Professor
Joe Raaymakers

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Document Summary

Remember that for small sample sizes, the test procedures require that the distribution of the observations should be approximately normally distributed. For larger sample sizes, the test procedures are appropriate regardless of the actual distribution of the observations due to the central limit theorem. If the sample size is small, the t-procedure generally provides sensible results unless the observations are clearly not normally distributed. The most convenient way to assess the precision afforded by a certain sample size is to estimate the length of the resulting two-sided confidence interval.

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Related Questions

A data set contains information from a random sample of 120 different female dogs who have bred,
40 of which are labradors, 40 are golden retrievers, and 40 are German shepherds.
The variables are:
• Breed: breed of the dog
• LitterSize: the number of pups in the dog’s most recent litter
The data are contained in the file dogs.sav.


(a) [6 marks] Is there evidence of a difference in the population mean litter sizes for the three
breeds? Be sure to state the null and alternative hypotheses, test statistic, null distribution,
p-value, decision, and an appropriate conclusion in plain language.


(b) [3 marks] If appropriate, perform post-hoc tests to determine which breeds have significantly
different mean litter sizes. If post-hoc tests are not appropriate, explain the purpose of a post-hoc
test and why it’s not appropriate in this example.


(c) [2 marks] Report the 95% pairwise confidence interval for the mean difference in litter sizes
between golden retrievers and German shepherds. Interpret this interval with a sentence in plain
language.


(d) [3 marks] What are the assumptions of the analysis performed in part (a)? State whether
each assumption is reasonable with reference to appropriate SPSS output.

 

Two random samples are taken, one from among first-year students and the other from among fourth-year students at a public university. Both samples are asked if they favor modifying the student Honor Code. A summary of the sample sizes and number of each group answering yes'' are given below:

 

First-Years (Pop. 1):𝑛1=91,𝑥1=55 Fourth-Years (Pop. 2):,𝑛2=100,𝑥2=62

 

Is there evidence, at an 𝛼=0.06 level of significance, to conclude that there is a difference in proportions between first-years and fourth-years? Carry out an appropriate hypothesis test, filling in the information requested.

A. The value of the standardized test statistic:

B. The rejection region for the standardized test statistic:

C. The p-value is:

 

Happy Clown Carnival Company is scheduled to appear in a city on a particular Saturday. The profits that will be obtained depend heavily upon the weather. If the weather is rainy, the carnival loses $15,000; if it is not rainy, the carnival makes a profit of $5,000. Historically, the probability of rain on any given day is 10%.

The carnival has to set up equipment for its show, but it can cancel the show prior to setting up its equipment. Cancelling the show results in a loss of $1,000, regardless of the weather. However, for an additional $250 cost, the carnival can postpone its setup decision until the Friday before the scheduled performance. At that time, the carnival can obtain the local weather forecast for Saturday’s weather.

The Weather Bureau has compiled data on its past forecasts. When it was rainy, the Weather Bureau had predicted rain 74% of the time. When it was not rainy, the Weather Bureau had predicted no rain 86% of the time.

(a) Set up the decision tree and determine the decision (SETUP or CANCEL) that maximizes the expected profit without waiting for a weather forecast on the blank page provided. Leave enough space to add to this tree in part (d).(15 pts for correct tree, 12 pts for decision and backup calculations.)

(b) Determine the Expected Value of Perfect Information (EVPI) for the situation depicted in your decision tree from (a).(7 pts.)

(c) Append to your decision tree in (a) a decision tree diagram that includes waiting for the results of the weather forecast. Be sure to clearly label the branches and nodes. (15 pts.)

(d) Determine the two posterior probability distributions for Saturday’s actual weather given the weather forecasts on Friday and put your results in the appropriate places on your tree in (c). (Hint: Bayes. ). (15 pts.)

(e) . Calculate the Expected Values at each appropriate node for the part of the tree diagram you added in (c). (30 pts.)

(f) Based on your results in (e), what is your optimal decision strategy? (5 pts.)

(g) Calculate the Expected Value of Sample Information (EVSI) and the efficiency of sample information for this situation. Also calculate the Expected Net Gain of Sample Information. (ENGSI is the amount you expect to gain from the sample information after accounting for what it cost you to get that information.) (5 pts.) Use this page for your decision tree. Please write clearly enough so I can tell what your answers are.