Question 1196534: A marketing research company is estimating which of two soft drinks college students prefer. A random sample of 100 college students produced the following 95% confidence interval for the proportion of college students who prefer drink A: (0.262, 0.622). What would happen to the confidence interval if the confidence level were changed to 90%?
There would be no change in the width of the interval.
The interval would get narrower.
The interval would get wider.
It is impossible to tell until the 90% interval is constructed
Answer by math_tutor2020(3817)   (Show Source):
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Answer: B) The interval would get narrower.
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Explanation:
The margin of error formula for proportions is
E = z*sqrt(phat*(1-phat)/n)
where,
z = critical value based on the confidence level
phat = sample proportion
n = sample size
We'll fix phat and n to be constant
They won't change in value.
Let's say phat = 0.50 and n = 100
At 95% confidence, the z critical value is roughly 1.96
At 90% confidence, the z critical value is roughly 1.645
Use a table like this
https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf
to get those values. Look at the bottom row labeled "Z".
Compute the value of E for the 95% confidence level
E = z*sqrt(phat*(1-phat)/n)
E = 1.96*sqrt(0.50*(1-0.50)/100)
E = 0.098
Now do the same for the 90% confidence level.
Keep phat and n the same from earlier.
E = z*sqrt(phat*(1-phat)/n)
E = 1.645*sqrt(0.50*(1-0.50)/100)
E = 0.08225
The margin of error has gone from 0.098 to 0.08225
The margin of error decreases when the confidence level decreases.
This makes the confidence interval get more narrow.
The confidence interval has the width of 2*E, so we can think of E as the radius of the confidence interval.
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Here's an alternative way to think about it:
Let's say your friend drew out a number line, and marked it from 1 to 10 including both endpoints.
Then your friend picked one of the numbers on that number line.
Your goal is to guess their number.
You are 100% confident it is somewhere between 1 and 10.
This is the largest confidence interval possible, and hence the widest width possible along the number line.
If you lowered the confidence down to 90%, then you could have a subinterval of say from 2 to 10, or something along those lines.
The friend's number may be in this range, but you aren't fully certain.
Then an 80% confidence interval could be going from 3 to 10 as one possible subinterval. The level of uncertainty has dropped further.
Each time the confidence decreases, the interval shrinks.
Similar question:
https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1196535.html
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