The Basics
What 3 conditions should be met to calculate a confidence interval?
Random
Independent (10%)
Normal (Large Counts)
Find z* for a 95% confidence level. Round to the hundredth.
1.96
What is the formula for calculating a confidence interval for a population proportion?
hat p pm z^ast sqrt ((hat p(1-hat p))/n)
What is the formula for calculating a confidence interval for a population mean?
bar x pm t^ast s/sqrt (n)
What do we always do with our answer when solving for n?
Round up
What calculator command do we use to find the confidence interval for a population proportion?
1-PropZint
Find z* for a 99% confidence level. Round to the hundredth.
2.58
At the Renaissance Festival, there are many people who are dressed up in various characters. You are interested in the true proportion of people who dress-up for the festival. You randomly select 300 people who enter the festival on a randomly selected Saturday of the festival, and record whether they are dressed in a costume. You find that 125 of people have costumes on.
Explain the difference between p and πΜ in the context of this problem?
p represents the true proportion of people who dress up in costume at the Renaissance Festival
πΜ = 125/300 = 0.417 represents the sample proportion of people who dress up in costume on the randomly selected day
biologist is studying the growth of a certain type of plant. She measures the heights (in cm) of 12 randomly selected plants after 8 weeks of growth:
24, 26, 23, 25, 27, 26, 24, 25, 30, 21, 25, 30
What are we trying to estimate by creating a confidence interval?
mu
= average amount of plant growth in cm
The population of the United States is approximately 330 million people. A research firm wants to determine the opinions of all Americans on a new public policy, with 90% confidence, and a margin of error of 2%. What is the appropriate number of people the firm should survey?
1692
What calculator command do we use when finding a confidence interval for a population mean if we do not know the population standard deviation?
TInterval
Find z* for a 92.5% confidence level. Round to the hundredth.
1.78
Find the standard error for πΜ=0.85 and n=10.
sqrt(((0.85)(0.15))/10))
ππΈπΜ=0.11
Find the standard error for:
π₯Μ =20
S = 3.5
n = 15
3.5/sqrt15
"ππΈ"=0.90
Find the sample size required to estimate the population mean.
A movie theatre owner wants to construct a 98% confidence interval to estimate the mean age of movie goers. They want the margin of error to be 1.5 years, and will use sigma = 19.5 based on reports from the Motion Picture Association of America. What is the minimum sample size needed?
918
A nutritionist wants to estimate the average amount of sugar in a certain brand of yogurt. She takes an SRS of 20 containers and measures the sugar content, in grams. She creates a 90% confidence interval for the true amount of sugar in this brand of yogurt: (11.42, 12.58). What is the point estimate?
12
Find t* for a 95% confidence level with a sample size of 25. Round to the hundredth.
2.06
Find the 93% confidence interval for the following:
πΜ=0.52
n = 25
0.52 pm 1.81(0.1)
(0.34, 0.70)
Find the 95% confidence interval for:
π₯Μ =92.5
S = 27.54
n = 12
92.5 pm 2.2(7.95)
(75, 110)
The manager for a radio station wants to estimate the mean length of all songs published after 2000. How many songs should be in the sample if he wants to be 90% confident, with a margin of error of 15 seconds? The manager will be using sigma = 35 seconds based on historical song data.
15
A nutritionist wants to estimate the average amount of sugar in a certain brand of yogurt. She takes an SRS of 20 containers and measures the sugar content, in grams. She creates a 90% confidence interval for the true amount of sugar in this brand of yogurt: (11.42, 12.58). What is the margin of error?
0.58
Find t* for a 99% confidence level with a sample size of 25. Round to the hundredth.
2.80
In a poll conducted among 350 human resource hiring officers, 44.6% of them said that face piercings and tattoos were hiring βred flagsβ. Construct a 99% confidence interval estimate of the proportion of all HR officers who believe that face piercings and tattoos were hiring βred flagsβ.
(0.37728, 0.51415)
A biologist is studying the growth of a certain type of plant. She measures the heights (in cm) of 12 randomly selected plants after 8 weeks of growth:
24, 26, 23, 25, 27, 26, 24, 25, 30, 21, 25, 30
Assuming all conditions are met, construct a 90% confidence interval.
(24.15, 26.85)
The Environmental Protection Agency wants to conduct a survey to gauge public proception of itsβ mission and vision. From previous studies, they know that they need to poll at least 300 people to achieve their desired level of confidence. If they want to keep the same level of confidence but divide the margin of error by one fourth, how many people will they have to poll?
4800