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Chapter 12: Problem 5

Determine the expected counts for each outcome. $$ \begin{array}{lllll} \hline \boldsymbol{n}=\mathbf{5 0 0} & & & & \\ \hline p_{i} & 0.2 & 0.1 & 0.45 & 0.25 \\ \hline \text { Expected counts } & & & & \\ \hline \end{array} $$

Short Answer

Expert verified
100, 50, 225, 125

Step by step solution

01

- Understand the Problem

You are given a total number of trials, represented by the parameter n, which in this case is 500. You are also provided with probabilities for each outcome, denoted as \( p_i \) for the various categories. Your task is to determine the expected count for each outcome.
02

- Formula for Expected Count

The formula to calculate the expected count for each outcome is given by \( E_i = n \times p_i \), where \( E_i \) is the expected count, \( n \) is the total number of trials, and \( p_i \) is the probability of the outcome.
03

- Calculate the Expected Counts

Using the formula from Step 2, you can calculate the expected counts for each outcome: For the first outcome: \( E_1 = 500 \times 0.2 = 100 \)For the second outcome: \( E_2 = 500 \times 0.1 = 50 \)For the third outcome: \( E_3 = 500 \times 0.45 = 225 \)For the fourth outcome: \( E_4 = 500 \times 0.25 = 125 \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

probability distribution
A probability distribution is a mathematical function that gives the probabilities of occurrence of different possible outcomes. For example, in our exercise, we have four possible outcomes with given probabilities: 0.2, 0.1, 0.45, and 0.25. Each number represents the chance of that particular outcome happening. The sum of all probabilities in a distribution always equals 1, showing that one of the possible outcomes will definitely occur.
expected count formula
The expected count formula helps to determine the average outcome expected over a large number of trials. The formula is: \( E_i = n \times p_i \), where:
  • \( E_i \) is the expected count for a specific outcome.
  • \( n \) is the total number of trials.
  • \( p_i \) is the probability of that outcome.
outcome analysis
Outcome analysis involves understanding and predicting outcomes based on given probabilities. From our exercise, using 500 trials and the given probabilities, we have calculated the expected counts for four different outcomes. For the first outcome, with a probability of 0.2, the expected count is 100. This analytic step is crucial in many fields, like quality control and market research.
multinomial distribution
The multinomial distribution is a generalization of the binomial distribution. It models the probabilities of obtaining a particular combination of counts for each category. In our case, we consider the probabilities for four different outcomes. Each outcome follows the multinomial distribution, represented by its own probability. This distribution helps to calculate the expected counts precisely and predict various scenarios in fields like genetics or polling data.
statistical calculation
Statistical calculation is the backbone of analyzing data. Using the steps outlined for our problem, we derive the expected counts by multiplying the total number of trials (500) by each outcome's probability. Performing these calculations:
  • For the first outcome: \( 500 \times 0.2 = 100 \)
  • For the second outcome: \( 500 \times 0.1 = 50 \)
  • For the third outcome: \( 500 \times 0.45 = 225 \)
  • For the fourth outcome: \( 500 \times 0.25 = 125 \)
Each step demonstrates the utility of statistical calculations in predicting and analyzing outcomes accurately.

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Most popular questions from this chapter

Does the location of your seat in a classroom play a role in attendance or grade? To answer this question, professors randomly assigned 400 students a general education physics course to one of four groups. The 100 students in group 1 sat 0 to 4 meters from the front of the class, the 100 students in group 2 sat 4 to 6.5 meters from the front, the 100 students in group 3 sat 6.5 to 9 meters from the front, and the 100 students in group 4 sat 9 to 12 meters from the front. (a) For the first half of the semester, the attendance for the whole class averaged \(83 \% .\) So, if there is no effect due to seat location, we would expect \(83 \%\) of students in each group to attend. The data show the attendance history for each group. How many students in each group attended, on average? Is there a significant difference among the groups in attendance patterns? Use the \(\alpha=0.05\) level of significance. (b) For the second half of the semester, the groups were rotated so that group 1 students moved to the back of class and group 4 students moved to the front. The same switch took place between groups 2 and \(3 .\) The attendance for the second half of the semester averaged \(80 \% .\) The data show the attendance records for the original groups (group 1 is now in back, group 2 is 6.5 to 9 meters from the front, and so on ). How many students in each group attended, on average? Is there a significant difference in attendance patterns? Use the \(\alpha=0.05\) level of significance. Do you find anything curious about these data? $$ \begin{array}{lllll} \hline \text { Group } & 1 & 2 & 3 & 4 \\ \hline \text { Attendance } & 0.84 & 0.81 & 0.78 & 0.76\\\ \hline \end{array} $$ (c) At the end of the semester, the proportion of students in the top \(20 \%\) of the class was determined. Of the students in group \(1,25 \%\) were in the top \(20 \%\); of the students in group \(2,21 \%\) were in the top \(20 \%\); of the students in group \(3,15 \%\) were in the top \(20 \%\); of the students in group \(4,19 \%\) were in the top \(20 \% .\) How many students would we expect to be in the top \(20 \%\) of the class if seat location plays no role in grades? Is there a significant difference in the number of students in the top \(20 \%\) of the class by group? (d) In earlier sections, we discussed results that were statistically significant, but did not have any practical significance. Discuss the practical significance of these results. In other words, given the choice, would you prefer sitting in the front or back?

test whether the population proportions differ at the \(\alpha=0.05\) level of significance by determining \((a)\) the null and alternative hypotheses, (b) the test statistic, (c) the critical value, and \((d)\) the P-value. Assume that the samples are dependent and were obtained randomly. $$ \begin{array}{cccc} & & {\text { Treatment A }} \\ & & \text { Success } & \text { Failure } \\ \hline{\text { Treatment B }} & \text { Success } & 84 & 21 \\ & \text { Failure } & 11 & 37 \\ \hline \end{array} $$

Celebrex Celebrex, a drug manufactured by Pfizer, Inc., is used to relieve symptoms associated with osteoarthritis and rheumatoid arthritis in adults. In clinical trials of the medication, some subjects reported dizziness as a side effect. The researchers wanted to discover whether the proportion of subjects taking Celebrex who reported dizziness as a side effect differed significantly from that for other treatment groups. The following data were collected. $$ \begin{array}{lrrrrr} & & {\text { Drug }} \\ \hline \text { Side Effect } & \text { Celebrex } & \text { Placebo } & \text { Naproxen } & \text { Diclofenac } & \text { Ibuprofen } \\ \hline \text { Dizziness } & 83 & 32 & 36 & 5 & 8 \\ \hline \text { No dizziness } & 4063 & 1832 & 1330 & 382 & 337 \end{array} $$ (a) Test whether the proportion of subjects within each treatment group who experienced dizziness are the same at the \(\alpha=0.01\) level of significance. (b) Construct a conditional distribution of side effect by treatment and draw a bar graph. Does this evidence support your conclusion in part (a)?

What's in a Word? In a recent survey conducted by the Pew Research Center, a random sample of adults 18 years of age or older living in the continental United States was asked their reaction to the word socialism. In addition, the individuals were asked to disclose which political party they most associate with. Results of the survey are given in the table. $$ \begin{array}{lccc} & \text { Democrat } & \text { Independent } & \text { Republican } \\ \hline \text { Positive } & 220 & 144 & 62 \\ \hline \text { Negative } & 279 & 410 & 351 \\ \hline \end{array} $$ (a) Explain why this data should be analyzed by homogeneity of proportions. (b) Does the evidence suggest individuals within each political affiliation react differently to the word socialism? Use the \(\alpha=0.05\) level of significance. (c) Construct a conditional distribution of reaction by political party. (d) Write a summary about the "partisan divide" regarding reaction to the word socialism.

A researcher wanted to determine whether bicycle deaths were uniformly distributed over the days of the week. She randomly selected 200 deaths that involved a bicycle, recorded the day of the week on which the death occurred, and obtained the following results (the data are based on information obtained from the Insurance Institute for Highway Safety). $$ \begin{array}{lc|lc} \begin{array}{l} \text { Day of } \\ \text { the Week } \end{array} & \text { Frequency } & \begin{array}{l} \text { Day of } \\ \text { the Week } \end{array} & \text { Frequency } \\ \hline \text { Sunday } & 16 & \text { Thursday } & 34 \\ \hline \text { Monday } & 35 & \text { Friday } & 41 \\ \hline \text { Tuesday } & 16 & \text { Saturday } & 30 \\ \hline \text { Wednesday } & 28 & & \\ \hline \end{array} $$ Is there reason to believe that bicycle fatalities occur with equal frequency with respect to day of the week at the \(\alpha=0.05\) level of significance?
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