Alcohol in Children’s Movies Listed below is a simple random s...

Question

Alcohol in Children’s Movies Listed below is a simple random sample of times (seconds) that animated children’s movies showed the use of alcohol (based on Data Set 20 “Alcohol and Tobacco in Movies” in Appendix B).

a. Are the requirements for constructing a 95% confidence interval estimate of the population mean satisfied? If so, construct that confidence interval.

Accepted Answer

All right, hello, everyone. So this question says, listed below is a sample random sample of the number of times caffeine, for example, coffee or energy drinks, was mentioned during episodes of a teen drama series. And here we have a data set of 10 values. Are the conditions met for constructing a 95% confidence interval estimate for the population mean number of caffeine mentions per episode? If yes, compute the 95% confidence interval for the meat. And here we have 4 different answer choices labeled A through D. So first and foremost, we have to consider the conditions to see if we even have to construct a confidence interval to begin with. First, the data set listed is a simple random sample. So that counts. Additionally, the sample size or lower case n happens to be less than 30, in this case specifically it's 10. Now, looking at the data values themselves, there aren't any extreme outliers that call your attention, and the numbers themselves appear to be fairly symmetric. Therefore, with all of this information in mind, conditions are satisfied and we can construct a confidence interval. So with that being said, let's begin with the sample mean or X bar. Now, X bar is equal to the sum of all of your data points divided by the sample size. So you would add up all of the data points you have. Which would give you a total of 53. And divide that by the total or the sample size of 10. 53 divided by 10 gives you a sample mean of 5.3. And from there you would proceed with the standard deviation. Now You can see on the left side of the screen here that I've already constructed a table with all of the data points already given to us. And the reason I did this is because recall that in order to find standard deviation, you have to take each data point. Subtract the data point from the sample mean, and then square the difference. The right side of the table represents the numbers that you would get from that calculation. So Starting off with the first point, for example, of 4, that would be 4 subtracted by the sample mean, and and then that difference is squared, which gives you 1.69. And from there you would repeat all of or you would repeat the calculation for the remaining data points. For 5, for example, that's 0.09. For 6, that's 0.49. For 3 it's 5.29. 7 is 2.89. For 5, it's once again 0.09. For 4 again it's 1.69. For 6, it's 0.49. For 8 it's 7.29. And for 5, it's 0.09. So now to calculate the standard deviation or As for the sample Standard deviation is equal to. The sum of all data points subtracted by the sample mean squared. Divided by The sample size subtracted by one. So going back to our table, you would go ahead and add together all of. The calculations made to fill out the right side of the table or the right column. That total comes out to 20.10. So going back to our standard deviation. Standard deviation will be equal to 20.10 divided by 10, subtracted by 1, which equals 9. Then evaluating this expression gives you a standard deviation of approximately 1.49. From here, you would then find the critical T value before finding the margin of error and then using the margin of error to find the confidence interval. In the calculation of the standard deviation, this value of N subtracted by 1 is equal to the degrees of freedom. So in this case, the degrees of freedom is equal to 9. For a 95% confidence interval. And To 9 degrees of freedom rather, T from the tea table is equal to 2.262. So now for the margin of error, recall that E is equal to that T value multiplied by the standard deviation, divided by the square root of the sample size. This equals 2.262. Multiplied by 1.49. Divided by square root of 10. And this equals approximately 1.1. So now it's with the margin of error. That you can then construct the values of the confidence interval. Because the confidence interval. Or CI Is equal to the sample mean added to and subtracted by. Capital E which represents the margin of error. So here that would be 5.3. Added to and subtracted by 1.1. Which ultimately gives you a confidence interval of 4.2. To 6.4. So going back to the answer choices given. In the question This would correspond to option D in the multiple choice, making that our correct answer. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

Key Concepts

Simple Random Sample

Confidence Interval

Normality Assumption

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