## Image Transcription for Educational Website ### Statistical Decision Making and Probability Analysis **Problem Context:** A city is evaluating a proposal to check the proportion of residences that have a certain detector. We consider \( p \) as the proportion of residences with detectors. To make a decision, a random sample of 25 residences is chosen. If the sample indicates that \( p < 0.80 \) (less than 80% of residences have detectors), the program will be implemented. Let \( x \) be the number of residences with a detector in the sample. The decision rule is to implement the program if \( x \leq 15 \). **Tasks:** **a.** What is the probability that the program is implemented when \( p = 0.80 \)? **b.** What is the probability that the program is not implemented if \( p = 0.70 \)? If \( p = 0.60 \)? **c.** How do the "error probabilities" of Parts (a) and (b) change if the value 15 in the decision rule is changed to 14? --- ### Voter Preferences Analysis **Scenario 7.58:** 90% of registered California voters favor banning the release of exit poll information in presidential elections until after the polls close. A sample of 25 voters is selected to decide on the ban's favorability. **Questions:** **a.** What is the probability that more than 20 favor the ban? **b.** What is the probability that at least 20 favor the ban? **c.** What are the mean value and standard deviation of the number of voters in the sample who favor the ban? **d.** If fewer than 20 voters favor the ban, is this inconsistent with the assertion that at least 90% favor the ban? Hint: Consider \( P(x < 20) \) when \( p = 0.9 \). --- ### Music Playlist Analysis **Scenario 7.59:** An MP3 playlist has 100 songs, out of which eight are by a specific artist. Songs are played randomly with replacement. The variable \( x \) represents the number of songs played until a song by the artist is played. (Note: The tasks related to this scenario were not fully visible in the image.) --- This transcription provides a detailed explanation of the probability and decision-making problems, suitable for an educational context. ### Chapter 7: Random Variables and Probability Distributions #### Exercises **b. Find the following probabilities:** i. \( P(4) \) ii. \( P(x \leq 4) \) iii. \( P(x > 4) \) iv. \( P(x \geq 4) \) **c. Interpret each of the probabilities in Part (b) and explain the difference between them.** --- **7.60** Sophie is a dog that loves to play catch. Unfortunately, she isn’t very good, and the probability that she catches a ball is only .1. Let \( x \) be the number of tosses required until Sophie catches a ball. a. Does \( x \) have a binomial or a geometric distribution? b. What is the probability that it will take exactly two tosses for Sophie to catch a ball? c. What is the probability that more than three tosses will be required? --- **7.61** Suppose that 5% of cereal boxes contain a prize and the other 95% contain the message, "Sorry, try again." Consider the random variable \( x \), where \( x \) = number of boxes purchased until a prize is found. --- **Note:** Bold exercises are answered in the back. A dataset is available online.