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Point Estimate
๐ฬ = (^) ๐๐ฅ ๐ฬ is the point estimate for the population proportion, ๐ฅ is the number of
successes, n is the sample size
๐ฬ = ๐๐๐๐๐ ๐ต๐๐ข๐๐+๐ฟ๐๐ค๐๐ ๐ต๐๐ข๐๐ 2 for the population proportion
๐ฅฬ
= ๐๐๐๐๐ ๐ต๐๐ข๐๐+๐ฟ๐๐ค๐๐ ๐ต๐๐ข๐๐ 2 for the population mean
๐ธ = ๐๐๐๐๐ ๐ต๐๐ข๐๐โ๐ฟ๐๐ค๐๐ ๐ต๐๐ข๐๐ 2 -> ๐ธ is the margin of error (can be used for either)
A survey was given to 1500 students in a school to find the proportion of students who are interested in taking a musical course. Determine the point estimate, margin of error, and number of students who responded they would be interested in a musical course, based on the given upper and lower bounds.
Lower bound = .682 Upper bound =.
Confidence Intervals
Population proportion: ๐ฬ ยฑ ๐ง๐ 2โ โ โ๐ฬ(1โ๐ฬ)๐ -> Use z-table
Population mean: ๐ฅฬ
ยฑ ๐ก๐ 2โ โ (^) โ๐๐ -> Use t-table
Population variance: (๐โ1)๐
2 ๐๐ 2^2 โ^ ,^
(๐โ1)๐ ^2 ๐1โ๐ 2^2 โ ->^ Use chi-square table
Population standard deviation: (^) โ
(๐โ1)๐ ^2 ๐๐ 2^2 โ^ , โ
(๐โ1)๐ ^2 ๐1โ๐ 2^2 โ ->^ Use chi-square table
Common critical values for the POPULATION PROPORTION ONLY:
For the population mean, variance, or standard deviation, the degrees of freedom (df), will also be needed to find the critical value.
๐๐ = ๐ โ 1
Sample Size
Always round up to the next integer
Population proportion: ๐ = ๐ฬ(1 โ ๐ฬ)(๐ง๐ 2 ๐ธโ )^2
**If there are no prior estimates, then use ๐ฬ =.
Population mean: ๐ = (
๐ง๐ 2โ โ๐
2
Examples: A survey of 500 airline passengers found that 338 were satisfied with the service they received from the flight attendants. Calculate and interpret a 95% confidence interval for the proportion of passengers who are satisfied with the service from flight attendants.
๏ท Because we are looking for a population proportion, first we need to find the point estimate, and then we will use the z-table in our confidence interval for the critical value.
๐ฬ =
Now, we use ๐ฬ ยฑ ๐ง๐ 2โ โ โ๐ฬ(1โ๐ฬ)๐ , and use the z-table. For a 95% confidence interval, the
critical value is 1.
Confidence Level
Critical Value (Z-Score) 90% 1. 95% 1. 98% 2. 99% 2.
A survey given to 25 incoming freshmen at a college in Illinois has shown that the average composite ACT score of an incoming freshman is 21.4. The standard deviation of these ACT scores for those freshmen is 5.78. Calculate and interpret a 90% confidence interval for the population standard deviation of composite ACT scores at that college in Illinois.
๏ท Because we are estimating the population standard deviation, we will use the chi- square table. For the confidence interval, we will use (^) โ(๐โ1)๐
2 ๐๐ 2^2 โ^ , โ
(๐โ1)๐ ^2 ๐1โ๐ 2^2 โ
๐ 2 =^
1โ. 2 = .05,^ 1 โ^
๐ 2 = 1 โ .05 =.
The degrees of freedom is 25-1, or 24. The two critical values then come from df=24, ๐/2 = .05, and df=24, ๐ = .95. Those critical values are 36.415 and 13.
(25 โ 1)5.78^2
(25 โ 1)5.78^2
Our 90% confidence interval is thusly (4.69, 7.61)
We are 90% confident that the population standard deviation of composite ACT scores at the college in Illinois is between 4.69 and 7.61.
Using the ACT data, compute and interpret a 99% confidence interval for the mean composite ACT score at the college in Illinois.
๏ท Because we are estimating the population mean, we will need to use the t-table. The degrees of freedom is still 24. Now we will find ๐ผ/2.
๐ 2 =^
1โ. 2 =.
With df=24, and ๐ = .005, we look at the t-table, and get a critical value of 2.
Now we use ๐ฅฬ
ยฑ ๐ก๐ 2โ โ (^) โ๐๐
The 99% confidence for the mean composite ACT score is (18.167, 24.633)
We are 99% confident that the population mean composite ACT score at the college in Illinois is between 18.167 and 24.
A political firm wants to conduct a survey to gauge voter interest of a particular candidate in the state of Minnesota. They are looking for a 95% confidence level, and the margin of error desired on the survey is 2%. What is the sample size needed if the prior estimate of people in favor of this candidate is 41%?
๏ท Because we are working with population proportion, we will use ๐ = ๐ฬ(1 โ ๐ฬ)(๐ง๐ 2 ๐ธโ )^2 ๏ท Also, ๐ง๐ 2โ = 1.96 for 95% confidence
2 = 2323.
So the political firm would need 2324 participants in the survey
What if there were no prior estimates?
๏ท Without prior estimates, we use ๐ฬ =.
๐ = .5(1 โ .5) (1.96.02 )
2 = 2401
With no prior estimates, the political firm would need 2401 participants for the survey
