Prayer in school A New York Times/CBS News Poll asked the...

Prayer in school A New York Times/CBS News Poll asked the question, “Do you favor an amendment to the Constitution that would permit organized prayer in public schools?” Sixty-six percent of the sample answered “Yes.” The article describing the poll says that it “is based on telephone interviews conducted from Sept. 13 to Sept. 18 with 1,664 adults around the United States, excluding Alaska and Hawaii. . . . The telephone numbers were formed by random digits, thus permitting access to both listed and unlisted residential numbers.” The article gives the margin of error for a 95% confidence level as 3 percentage points. (a) Explain what the margin of error means to someone who knows little statistics. (b) State and interpret the 95% confidence interval. (c) Interpret the confidence level.

Prayer in school A New York Times/CBS News Poll asked the question, “Do you favor an amendment to the Constitution that would permit organized prayer in public schools?” Sixty-six percent of the sample answered “Yes.” The article describing the poll says that it “is based on telephone interviews conducted from Sept. 13 to Sept. 18 with 1,664 adults around the United States, excluding Alaska and Hawaii. . . . The telephone numbers were formed by random digits, thus permitting access to both listed and unlisted residential numbers.” The article gives the margin of error for a 95% confidence level as 3 percentage points.
(a) Explain what the margin of error means to someone who knows little statistics.
(b) State and interpret the 95% confidence interval.
(c) Interpret the confidence level.

Key Concepts

Confidence_Interval

A confidence interval is a range of values, derived from a statistic, that is likely to contain the true parameter of the population. It is constructed by taking the sample estimate and expanding it by the margin of error. For example, if a sample survey gives a point estimate and a margin of error, the confidence interval provides a lower and upper bound around that estimate, indicating that the true population value is likely to lie within this span at a specified level of confidence.

Confidence_Level

The confidence level represents the percentage of all possible samples that can be expected to include the true population parameter if the same survey were repeated multiple times. A 95% confidence level means that if the sampling process were repeated many times, approximately 95% of the calculated confidence intervals would contain the true parameter. It is a measure of how sure we are about the interval estimate constructed from the sample data.

Margin_of_Error

The margin of error is a statistical measure that indicates the range around the sample estimate within which the true value for the entire population is likely to fall. It quantifies the uncertainty associated with the sample estimate and reflects potential sampling variability. A smaller margin of error indicates more precision in the estimation, while a larger one suggests less certainty about how close the sample estimate is to the actual population value.

Transcript

00:02 Sample gave a proportion of 66%.

00:04 We know that that came from a sample size of 1 ,664 people.

00:10 And they tell us that the margin of air for this particular setting is 3 % if we're finding a 95 % and i'll just call it confidence interval.

00:25 And so to begin with, we want to look at what do we mean by that margin of air.

00:30 If we say the margin of air is 3%, what does that mean? and that means we're going to take some type of an estimate, and we know that when we get that estimate, which in our case is this 66%, we know that we are very doubtful that that's the exact answer.

00:48 So we add on and subtract away some type of a margin of air.

00:54 And that margin of air is going to be controlled by how big of a sample and by the level of confidence that we want to have.

01:01 So basically, i know this answer is not exactly right.

01:06 But based on my sample size and how confident i want to be, i know that if i subtract away this margin of error from my estimate and add on that error, i'll get some interval.

01:19 And this is what the population is, is our parameter.

01:23 The thing we're trying to find.

01:25 And i can have a low estimate and a high estimate, estimate and i can have a certain level of confidence that once i take my estimate plus or minus this margin of error, that my actual proportion is somewhere in between these numbers.

01:40 So again, basically that margin of error helps us look at what would we think that interval or that value that we're trying to find is because we're doubtful that it's exactly that estimate.

01:52 Now, how do you calculate the confidence interval? our confidence interval would be that 66 % plus or minus the margin of air.

02:06 And so if we subtract away, we would find that that's equivalent to 63 % to 69%...

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