Using the information from the One Proportion applet shown below, determine a 25D confidence interval for the population parameter Probability of heads: 10,5 P Summary Stats Number of tosses. Number of repetitions (1000 Nean 0.500 SD 0.033 Animate Draw Samples Total = 1000 Proportion of heads b. Write a sentence to interpret the confidence interval in this context c Do we have evidence that a majority of US cell phone users age 16 to 17 years have talked on the phone while driving? Explain
The Correct Answer and Explanation is:
Based on the One Proportion Applet and the given information:
- Number of tosses per sample: 10
- Number of repetitions: 1000
- Mean proportion of heads: 0.500
- Standard deviation (SD): 0.033
a. 25% Confidence Interval for the Probability of Heads
To construct a 25% confidence interval, we use the formula for a confidence interval for a proportion: CI=p^±z∗⋅SE\text{CI} = \hat{p} \pm z^* \cdot \text{SE}
Where:
- p^=0.500\hat{p} = 0.500 (sample proportion)
- z∗z^* for a 25% confidence level ≈ 0.674 (since 25% confidence leaves 12.5% in each tail of the standard normal distribution)
- SE=0.033\text{SE} = 0.033 (standard deviation given)
Margin of Error=0.674⋅0.033≈0.0222\text{Margin of Error} = 0.674 \cdot 0.033 \approx 0.0222 CI=0.500±0.0222=(0.4778,0.5222)\text{CI} = 0.500 \pm 0.0222 = (0.4778, 0.5222)
b. Interpretation of the Confidence Interval
We are 25% confident that the true probability of getting heads when flipping a fair coin lies between 0.4778 and 0.5222. This means that in 25% of all possible random samples of size 10, the interval computed this way will contain the true proportion of heads.
c. Is there evidence that a majority of US cell phone users age 16 to 17 have talked on the phone while driving?
No, this coin-toss simulation does not provide any evidence related to US teen drivers. The One Proportion Applet and the coin toss simulation model a binomial experiment under the assumption of a fair coin (with true probability 0.5) and is meant to illustrate concepts of sampling variability and confidence intervals in general.
To determine if a majority (i.e., more than 50%) of US cell phone users aged 16 to 17 have talked on the phone while driving, you would need actual survey or observational data from that population. Once you have that data, you could calculate a sample proportion (e.g., 0.55), compute a confidence interval around it, and see whether the interval lies entirely above 0.5, which would suggest that a majority is likely.
Since the given simulation relates only to coin tosses, it doesn’t support a conclusion about teen driving behavior.
Summary
The One Proportion Applet simulates a binomial experiment using coin tosses. In this case, 1000 repetitions of tossing a fair coin 10 times were conducted. The average proportion of heads across all simulations was 0.500, with a standard deviation of 0.033. To construct a 25% confidence interval for the population parameter (the true probability of heads), we use the formula: CI=p^±z∗⋅SE\text{CI} = \hat{p} \pm z^* \cdot \text{SE}
Using z∗=0.674z^* = 0.674 for a 25% confidence level and the given SD, we find a margin of error of approximately 0.0222. Therefore, the confidence interval is (0.4778, 0.5222).
This interval means we are 25% confident that the true proportion of heads for a fair coin lies between 47.78% and 52.22%. It highlights the concept of sampling variability—that different random samples give different sample proportions, but most will fall near the true population value.
Regarding the question of whether a majority of US cell phone users aged 16 to 17 talk on the phone while driving, the coin toss simulation provides no relevant evidence. It was only a theoretical model to demonstrate statistical inference concepts. To assess teen driving behaviors, actual data from that specific population is needed. Then, by analyzing the sample proportion and confidence interval based on that data, one could determine whether a majority (more than 50%) is likely. Since no such data is presented here, we cannot conclude anything about teen driving behavior from the simulation alone.
