5. A theater owner wants to know how popular a movie is on the first (opening) day itis shown. therbeqe as they exit. -After the noon showing she asks 50 people randomly chosen, "Would you recommend this movie to a friend or family member?" Thirty people say, "Yes," " -At the \( 3: 00 \) showing she asks 50 random people and 22 say, "Yes." "At the \( 7: 00 \) final showing of the day, she asks 50 random people and 23 say, "Yes." Using this data, out of the 900 viewers how many people do you think will recommend the movie? How did you decide? How far off do you think your prediction might be?

To estimate how many of the 900 viewers will recommend the movie, we'll analyze the data collected from the three showings and make a prediction based on the overall response rate. ### **Data Summary:** 1. **Noon Showing:** - Sample Size: 50 - "Yes" Responses: 30 - **Proportion:** \( \frac{30}{50} = 0.60 \) (60%) 2. **3:00 PM Showing:** - Sample Size: 50 - "Yes" Responses: 22 - **Proportion:** \( \frac{22}{50} = 0.44 \) (44%) 3. **7:00 PM Showing:** - Sample Size: 50 - "Yes" Responses: 23 - **Proportion:** \( \frac{23}{50} = 0.46 \) (46%) ### **Overall Analysis:** - **Total Sample Size:** \( 50 + 50 + 50 = 150 \) viewers - **Total "Yes" Responses:** \( 30 + 22 + 23 = 75 \) viewers - **Overall Proportion of "Yes" Responses:** \( \frac{75}{150} = 0.50 \) (50%) ### **Prediction for 900 Viewers:** Using the overall proportion: \[ \text{Estimated "Yes" Responses} = 900 \times 0.50 = 450 \] **Thus, approximately **450 out of 900** viewers are expected to recommend the movie.** ### **Decision Reasoning:** - **Sampling Method:** The theater owner sampled an equal number of viewers (50) from each of the three showings, which provides a balanced view across different times of the day. - **Aggregation of Data:** By combining the "Yes" responses from all showings, we obtain a reliable overall proportion (50%) to base our prediction on. ### **Potential Prediction Error:** - **Sampling Variability:** Since the sample size is relatively small (150 out of 900), there's inherent sampling variability. The true proportion could reasonably vary. - **Confidence Interval:** A 95% confidence interval for the proportion can be calculated to understand the range of possible outcomes. For a proportion of 0.50 with a sample size of 150: \[ \text{Standard Error} = \sqrt{\frac{0.5 \times 0.5}{150}} \approx 0.0408 \] \[ \text{95% Confidence Interval} = 0.50 \pm (1.96 \times 0.0408) \approx [0.42, 0.58] \] - **Interpretation:** This means the actual number of "Yes" recommendations could reasonably range from: \[ 900 \times 0.42 = 378 \quad \text{to} \quad 900 \times 0.58 = 522 \] **Therefore, while the best estimate is 450 recommendations, the actual number could reasonably be between approximately 378 and 522 due to sampling variability.** ### **Conclusion:** Based on the sampled data, it's reasonable to predict that around **450 out of 900 viewers** will recommend the movie. However, due to natural sampling variability, the actual number could reasonably be somewhat higher or lower, potentially ranging between **378 and 522** recommendations.