In a sample of 200 people, 154 said they watched educational television. Find the 90% confidence interval of the true proportion of people who watch educational television. If the television company wanted to publicize the proportion of viewers, do you think it should use the 90% confidence interval?
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To find the confidence interval for the true proportion of people who watch educational television, we can use the formula:
Confidence Interval = Sample Proportion +/- Margin of Error
First, let's calculate the sample proportion:
Sample Proportion = Number of people who watch educational television / Total sample size
Sample Proportion = 154 / 200 = 0.77
Next, we need to calculate the Margin of Error. For a 90% confidence level, we can use the formula:
Margin of Error = Critical Value * Standard Error
The critical value depends on the desired confidence level and the sample size. Since we have a sample size of 200 and want a 90% confidence level, we need to find the critical value for a 95% confidence level (the remaining 5% is divided equally between the two tails of the distribution).
Looking up the critical value in the Z-table, we find that it is approximately 1.645.
Standard Error = ā((Sample Proportion * (1 - Sample Proportion)) / Sample Size)
Standard Error = ā((0.77 * (1 - 0.77)) / 200) = 0.0306
Margin of Error = 1.645 * 0.0306 = 0.0503 (rounded to four decimal places)
Now we can calculate the confidence interval:
Confidence Interval = Sample Proportion +/- Margin of Error
Confidence Interval = 0.77 +/- 0.0503
Confidence Interval = (0.7197, 0.8203)
So, the 90% confidence interval for the true proportion of people who watch educational television is approximately (0.72, 0.82).
Now, whether the television company should use this confidence interval to publicize the proportion of viewers depends on their specific goals. A 90% confidence interval means that there is a 90% chance that the true proportion falls within the interval. It provides a range estimate of the true proportion, indicating the level of uncertainty.
If the company wants to present a conservative estimate and inform viewers that the actual proportion of viewers is likely to be within this range, then using the 90% confidence interval would be appropriate. However, if the company wants to present a more precise estimate, they may consider using a smaller confidence interval, such as a 95% or 99% confidence level, which would provide a narrower range but with a higher level of certainty.