## Educational Website Transcription### OverviewThe image displays an exercise related to confidence intervals. It involves simulating 19 samples of size $ n = 120 $ from a population. The table and the accompanying graphs illustrate the 75% and 95% confidence intervals for these samples.### TableThe table shows the lower and upper limits for both 75% and 95% confidence intervals for each sample (S1 to S20). Here are a few examples:- **S1:** - 75% CI: Lower limit = 0.66, Upper limit = 0.76 - 95% CI: Lower limit = 0.61, Upper limit = 0.77- **S2:** - 75% CI: Lower limit = 0.64, Upper limit = 0.73 - 95% CI: Lower limit = 0.60, Upper limit = 0.76- **S20:** - 75% CI: Lower limit = 0.69, Upper limit = 0.77 - 95% CI: Lower limit = 0.66, Upper limit = 0.82### GraphsThe graphs illustrate the confidence intervals visually:1. **75% Confidence Intervals:** Displayed on the left, showing shorter ranges compared to the 95% intervals.2. **95% Confidence Intervals:** Displayed on the right, indicating wider ranges to capture the population proportion with higher certainty.### Multiple Choice Questions#### Part (c)The exercise asks the reader to choose the correct statement about 95% confidence intervals:- Highlighting that while constructing these intervals for 20 samples, the percentage containing the population proportion should be around 95%, but not exactly.#### Part (d)This section lists statements requiring the reader to select all true options:- Discusses the relationship between confidence level and the probability of containing the population proportion.- Considers the width of the interval in relation to the sample size and confidence level.### Learning Points- **Understanding Confidence Intervals:** Illustrates the concept of confidence intervals, emphasizing their use in estimating population proportions.- **Effect of Confidence Levels on Intervals:** Shows how higher confidence levels result in wider intervals.- **Practical Application:** Provides a hands-on approach to understanding confidence intervals through simulation and visual data representation. You are looking at a population and are interested in the proportion $ p $ that has a certain characteristic. Unknown to you, this population proportion is $ p = 0.70 $. You have taken a random sample of size $ n = 120 $ from the population and found that the proportion of the sample that has the characteristic is $ \hat{p} = 0.76 $. Your sample is Sample 1 in the table below.(a) Based on Sample 1, graph the 75% and 95% confidence intervals for the population proportion. Use 1.150 for the critical value for the 75% confidence interval, and use 1.960 for the critical value for the 95% confidence interval.- Enter the lower and upper limits on the graphs to show each confidence interval. Write your answers with two decimal places.- For the points (● and ⊗), enter the population proportion, 0.70.**Graphs:**- **75% confidence interval**: The line ranges from 0.69 to 0.83 with the center point at 0.76 marked.- **95% confidence interval**: The line ranges from 0.54 to 0.85 with the center point at 0.69 marked.(b) Press the "Generate Samples" button to simulate taking 19 more samples of size $ n = 120 $ from the same population. The confidence intervals for these samples are drawn automatically. Complete parts (c) and (d) below the table.**Table:**- Includes samples S1 to S20 with columns for the 75% and 95% confidence interval lower and upper limits.**Confidence Intervals Graphs:**- Diagrams on the right show bars for each of 20 samples’ 75% and 95% confidence intervals.(c) Notice that for $\frac{20}{20} = 100\%$ of the samples, the 95% confidence interval contains the population proportion. Choose the correct statement:- When constructing 95% confidence intervals for 20 samples of the same size from the population, exactly 95% of the samples must contain the population proportion. There must have been an error with the way our samples were chosen.- When constructing 95% confidence intervals for 20 samples of the same size from the population, the percentage of the samples that contain the population proportion should be

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You are looking at a population and are interested in the proportion p that has a certain characteristic. Unknown to you, this population proportion is p=0.70. You have taken a random sample of size = 120 from the population and found that the proportion of the sample that has the characteristic is p-0.76. Your sample is Sample 1 in the table below. (In the table, Sample 1 is written "S1", Sample 2 is written "S2", etc.) (a) Based on Sample 1, graph the 75% and 95% confidence intervals for the population proportion. Use 1.150 for the critical value for the 75% confidence interval, and use 1.960 for the critical value for the 95% confidence interval. (If necessary, consult a list of formulas.) • Enter the lower and upper limits on the graphs to show each confidence interval. Write your answers with two decimal places. For the points ( and ), enter the population proportion, 0.70. 0.54 0.54 0.54 0.54 75% confidence interval 0.69 P X 95% confidence interval 0.69 X 75% 75% 95% 95% lower upper lower upper limit limit limit limit $1 0.76 ? $2 0.69 0.64 $3 0.68 0.63 ? ? ? 0.74 0.61 0.77 0.73 0.60 0.76 S4 0.63 0.58 0.68 0.54 0.72 S5 0.67 0.62 0.72 0.59 0.75 S6 0.67 0.62 0.72 0.59 0.75 $7 0.72 0.67 0.77 0.64 0.80 S8 0.73 0.68 0.78 0.65 0.81 59 0.63 0.58 0.68 0.54 0.72 $10 0.71 0.66 0.76 0.63 0.79 $11 0.69 0.64 0.74 0.61 0.77 $12 0.71 0.66 0.76 0.63 0.79 $13 0.70 0.65 0.75 0.62 0.78 $14 0.72 0.67 0.77 0.64 0.80 $15 0.70 0.65 0.75 0.62 0.78 $16 0.76 0.72 0.80 0.68 0.84 $17 0.71 0.66 0.76 0.63 0.79 $18 0.72 0.67 0.77 0.64 0.80 $19 0.67 0.62 0.72 0.59 0.75 $20 0.74 0.69 0.79 0.66 0.82 $ 0.54 5 0.85 0.85 (b) Press the "Generate Samples" button below to simulate taking 19 more samples of size 120 from the same population. Notice that the confidence intervals for these samples are drawn automatically. Then complete parts (c) and (d) below the table. 0.85 0.85 75% confidence intervals 0.85 0.54 20 (c) Notice that for 100% of the samples, the 95% confidence interval contains 20 the population proportion. Choose the correct statement. O When constructing 95% confidence intervals for 20 samples of the same size from the population, exactly 95% of the samples must contain the population proportion. There must have been an error with the way our samples were chosen. O When constructing 95% confidence intervals for 20 samples of the same size from the population, the percentage of the samples that contain the population proportion should be close to 95%, but it may not be exactly 95%. 95% confidence intervals F H H X H H S 0.85