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 90% confidence interval of the true proportion of people who watch educational television, we can use the formula:
Confidence Interval = Sample Proportion ± Margin of Error
Step 1: Calculate the Sample Proportion:
The sample proportion (p̂) is the number of people who watched educational television divided by the total sample size. In this case, 154 out of 200 people watched educational television, so the sample proportion is:
p̂ = 154/200 = 0.77
Step 2: Calculate the Margin of Error:
The margin of error represents the range within which we can be confident the true population proportion lies. It is calculated using the formula:
Margin of Error = Critical Value * Standard Error
The critical value is obtained from the standard normal distribution table or using statistical software. For a 90% confidence level, the critical value is approximately 1.645.
To calculate the standard error, we use the formula:
Standard Error = √( (p̂ * (1 - p̂)) / n )
where p̂ is the sample proportion and n is the sample size.
Plugging in the values, we get:
Standard Error = √( (0.77 * (1 - 0.77)) / 200 ) ≈ 0.028
Now, calculating the margin of error:
Margin of Error = 1.645 * 0.028 ≈ 0.046
Step 3: Calculate the Confidence Interval:
Finally, we can calculate the confidence interval using the sample proportion and the margin of error:
Confidence Interval = p̂ ± Margin of Error
Confidence Interval = 0.77 ± 0.046
The 90% confidence interval for the true proportion of people who watch educational television is approximately (0.724, 0.816).
Now, should the television company use this 90% confidence interval to publicize the proportion of viewers? The answer depends on the level of confidence desired for the claim. A 90% confidence interval means that if we repeated this study many times and constructed a confidence interval for each study, approximately 90% of those intervals would contain the true population proportion. If the television company wants a higher level of confidence, say 95% or 99%, a wider confidence interval would be appropriate. However, if a 90% confidence level is sufficient, then it could be used to publicize the proportion of viewers.