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?
To find the 90% confidence interval for the proportion of people who watch educational television, we can use the formula:
Confidence interval = sample proportion ± (critical value) * (standard error)
Let's break it down step by step:
1. Calculate the sample proportion:
Sample proportion (p̂) = number of people who watch educational television / total sample size
In this case, the number of people who watch educational television is 154 and the total sample size is 200. So, the sample proportion is:
p̂ = 154 / 200 = 0.77
2. Calculate the standard error:
Standard error = √((p̂ * (1 - p̂)) / n)
In this case, p̂ is 0.77 (from step 1) and n is the total sample size, which is 200. So, the standard error is:
Standard error = √((0.77 * (1 - 0.77)) / 200) ≈ 0.026
3. Determine the critical value:
The critical value is based on the desired confidence level. For a 90% confidence level, we need to find the Z-score that leaves 5% area in the tails (since we want a two-tailed test). We can look up this value in the standard normal distribution table or use a statistical software. The critical value for a 90% confidence level is approximately 1.645.
4. Calculate the confidence interval:
Confidence interval = p̂ ± (critical value) * (standard error)
Using the values we have calculated:
Confidence interval = 0.77 ± 1.645 * 0.026
≈ 0.77 ± 0.043
The 90% confidence interval for the true proportion of people who watch educational television is approximately (0.727, 0.813).
To determine whether the television company should use this interval to publicize the proportion of viewers, we need to consider the level of certainty desired. A 90% confidence interval means that we can be 90% confident that the true proportion of people who watch educational television falls within the given interval.
If the television company wants to present a more conservative estimate and be more certain about their claims, they may prefer a higher confidence level like 95% or 99%. On the other hand, if they are willing to accept some level of uncertainty and want to reach a wider audience, a 90% confidence interval would suffice.
Ultimately, the decision depends on the specific circumstances and considerations of the television company.