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Statistical Sciences 2141A/B Lecture 5: Section 1.2 and 1.3

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

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Document Summary

A sample space (or state space), s, of an experiment is the set of all possible experimental outcomes. An event is any collection of outcomes contained in the sample space, s. An event is simple if it consists of exactly one outcome. An event is compound if it consists of more than one outcome. We take a collection of solutions contained in that space. If that collection has only one solution, we call it "simple". If that collection has more than one solution, we call it "compound" An event is just a collection of solutions, and can thus be considered a data set. So relationships and results that we studied in section 6. 3 about data sets can be applied. The union of two events a and b is the event consisting of (all) outcomes that are in either event a or in event b or in both events. This is denoted by a b (read "a or b")

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Related Questions

Never forget that even small effects can be statistically significant if the samples are large. During a three-year period, 8 of the 57 headed by men and 6 of the 35 headed by women failed.

(a) Find the proportions of failures for businesses headed by women and businesses headed by men. These sample proportions are quite close to each other. Give the p-value for the hypothesis that the same proportion of women's and men's businesses fail. (Use two-sided alternative). What can we conclude (use alpha as 0.05)?

The p-value was ___________ so we conclude that ____________.

 

(b) Now suppose that the same sample proportion came from a sample 30 times as large. That is 180 out of 1050 businesses headed by women and 240 out of 1710 businesses headed by men fail. Verify that the proportions of failures are exactly the same as in part (a). Repeat the test for the new data. What can we conclude?

The p-value was ___________ so we conclude that ____________.

c) It is wise to use a confidence interval to estimate the size of an effect rather than just giving a p-value. Give 95% confidence intervals for the difference between proportions of men's and women's businesses that fail for the settings of both (a) and (b).

Interval for smaller samples: ___________ to ___________.

Interval for larger samples: ___________ to ___________.

What is the effect of larger samples on the confidence interval?

 

One marketing strategy used by several companies is to send personalized emails or newsletters to their customers to encourage them to buy their product. However, the opening rate of these emails is very low when the targeted customers do not feel concerned by the email. This article 1 (https://www.crop.ca/en/blog/2019/258/) will allow us to answer the following questions. Here’s an excerpt from the article :

« A quick Google search for email and newsletter opening rates leads to dozens of sources, with rates ranging from 15% to 45%. Some even claim that opening rates higher than 25% are a resounding success. But seriously, can brands afford such poor performances ? And most importantly, why are opening rates so low ?

We asked the main stakeholders : consumers. 2,189 Canadians were ques- tioned about their attitudes and behaviours regarding branded newsletters and emails. The results follow. Of all branded communications received personally, Canadians report an average opening rate of 45%. »

(a) Do the results from the survey in the article in reference allow us to affirm that on average, Canadian consumers open more than 25 % of business communications sent by email and newsletters ? Suppose that the standard deviation in the population is 20 %. Use a significance level of α = 0, 05.

(b) From what average sample rate do we reject the null hypothesis of the test carried out in (a) ?

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.