Probability Analysis and Chi-Squared Test for Game Show Scenarios

Google Doc Access Directions: ● Please click on File in the upper left corner. ● If you are working on a Chromebook or Google Docs, choose the Make a copy option and save a copy of the document to your Google Drive. ● If not, choose the Download as option and then the Microsoft Word (.docx) option to download an editable copy of the document to your computer. Module Twelve Lesson One Assignment For this practice assignment, you need to answer the following questions to the best of your ability in the text box provided. Make sure that it is neatly organized so that your teacher can grade each specific step for full credit. Contestants on a game show spin a wheel like the one shown below. Each of the four outcomes on this wheel is equally likely and outcomes are independent from one spin to the next. image made by Jonna Ramsey ● The contestant spins the wheel. ● If the result is a skunk, no money is won and the contestant’s turn is finished. ● If the result is a number, the corresponding amount in dollars is won. The contestant can then stop with those winnings or can choose to spin again, and his or her turn continues. ● If the contestant spins again and the result is a skunk, all of the money earned on that turn is lost and the turn ends.

● The contestant may continue adding to his or her winnings until he or she chooses to stop or until a spin results in a skunk. 1. What is the probability that the result will be a number on all of the first three spins of the wheel? ¾*¾*¾=27/64 2. Suppose a contestant has earned $800 on his or her first three spins and chooses to spin the wheel again. What is the expected value of his or her total winnings for the four spins? e(4th win)= -800(¼)+100(¼)+500(¼)+200(¼) e(4th win)= 0 e(total win)= $800 3. A contestant who lost at this game alleges that the wheel is not fair. In order to check on the fairness of the wheel, that data in the table below were collected for 100 spins of this wheel. Result Skunk $100 $200 $500 Frequency 33 21 20 26 Based on these data, can you conclude that the four outcomes on this wheel are not equally likely? Give appropriate statistical evidence to support your answer. H0: The wheel is fair H1: The wheel is not fair In the case of a fair wheel, the expected frequency of occurrence of each type is 20. So, we have to perform the Chi-square test for goodness of fit. Number of groups n=4 2

Degrees of freedom v=n-1=3 P Value= 0.2366875 We cannot reject our null hypothesis. Hence, based on given data we can conclude that there is no significant evidence about the fact that the wheel is not fair. 4. When calculating the chi-squared test statistic, what happens when the observed values are exactly the same as our expected value? In such case, x^2=0 and so p-value=1 suggesting no significant evidence about the fact that the wheel is not fair. 5. When calculating the chi-squared test statistic, what happens when the observed values are very far away from our expected value? Large value of x^2 indicates very small p-value and in such case (p- value)<a (level of significance) suggesting significant evidence about the fact that the wheel is not fair. 6. What does a large chi-squared value mean when it comes to the p-value and the null hypothesis? A large chi-squared value leads to a small p-value. If the p-value is less than the chosen significance level (e.g., 0.05), you would reject the null hypothesis. This implies that there is significant evidence that the observed distribution differs from the expected distribution. 7. What does a small chi-squared value mean when it comes to the p-value and the null hypothesis? A small chi-squared value leads to a larger p-value. If the p-value is greater than the chosen significance level, you would fail to reject the 3

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null hypothesis. This suggests that there is not enough evidence to conclude a significant difference between the observed and expected distributions. 4

Copy of Module Twelve Lesson One Assignment (2) (1)

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