Vocab
This can be found by dividing the population standard deviation by the square root of the sample size.
Standard Error
What statistic MUST be present in order for me to use a Z-distribution to form a confidence interval?
Population standard deviation
If (99.8, 104.6) is a 90% confidence interval for a sample of basketball fifteen scores, what can you say about the average scores of the population?
That you are 90% sure your interval contains the population mean.
What is the critical Z-value for a 95% confidence interval
1.96
This is the singular best guess at estimating a population parameter.
Point Estimate
Which of these changes when I use a t-distribution as opposed to a Z-distribution: Sample size, confidence level, or critical value?
critical value
If the standard error of a sample is 2.3 and the critical value for a confidence level is 3.456, what is the margin of error?
7.9488
A 99% confidence interval was created to estimate a population proportion. The ME for the interval was 10 units wide. How could we revise this study in a way that would produce a narrower confidence interval for the population proportion?
We could either lower the CL% from 99% to something smaller (90% perhaps) OR we could increase the sample size and collect more data in our sample.
If my sample size is 12, how many degrees of freedom are there?
11
This is built around a point estimate in an effort to better approximate a population parameter.
Confidence Interval
Should I use a t-distribution or a z-distribution? Suppose that you want to find out the average weight of all players on the football team at a particular College. You are able to select ten players at random and weigh them. The mean weight of the sample of players is 198 lbs, with a standard deviation of 11.50 lbs.
t distribution
Find the Margin of Error for a 95% confidence interval with a standard error of 4.2 using a z-interval.
8.232
The average heights of a random sample of 400 people from a city is 1.75 m. It is known that the heights of the population are random variables that follow a normal distribution with a standard deviation of 0.4 m. Determine the interval of 95% confidence for the average heights of the population.
n = 400 x = 1.75 σ = 0.4 c = 0.95 z* = 1.96 (1.75 ± 1.96 • 0.4/20 ) → (1.7108, 1.7892)
If my t-value is 2.518, what is the required confidence level and sample size?
C=0.98 or 98%; n=22 (since df = 21)
This is added and subtracted from the point estimate in order to construct a confidence interval
Margin of Error
Should I use a t-distribution or Z? The average heights of a random sample of 400 people from a city is 1.75 m. It is known that the heights of the population are random variables that follow a normal distribution with a standard deviation of 0.4 m. Determine the interval of 95% confidence for the average heights of the population.
Z distribution
What is the margin of error for a sample with a critical value of 1.76, a sample size of 64, and a sample standard deviation of 3.5?
0.77
Suppose that you want to find out the average weight of all players on the football team at a particular College. You are able to select ten players at random and weigh them. The mean weight of the sample of players is 198 lbs, with a standard deviation of 11.50 lbs. Construct a 90 percent confidence interval for the population weight, if you presume the players' weights are normally distributed and state a conclusion.
We are 90 percent confident that the true population mean of football player weights is between 192 and 204 pounds.
Why are confidence intervals necessary?
Because the point estimate is not a very good approximation for the population parameter.
This is the number of free choices allowed by my parameter.
Degrees of freedom
I am estimating the mean of a population with an unknown distribution (and unknown standard deviation). My sample size is 15. What should I confirm before I create a t-interval?
I should check my sample data for outliers and strong skewness. If I have either then I should not use this methodology.
If the margin of error of a sample is 3.45, the sample size if 16, and the critical value for the confidence level is 1.876, then what is sample standard deviation?
7.356
If a sample of size n = 36 has a mean of 29 and a standard deviation of 4.3, find a 90% confidence interval for the population mean. Assume we do not know the population standard deviation.
df = 35 so use line with df=30 of Table b for t* value = 1.697
CI = (27.784, 30.216)
or CI = 29 +- 1.216
DAILY DOUBLE: Describe the 3 conditions that we need to check before creating a confidence interval.
1) Random data - use an SRS...
2) 10% rule: n <= 10%N and
3) Normal/Large Counts test
Proportions: np>=10 and n(1-p) >=10
Means: Population is Normal, or n >= 30 or check the sample data for outliers/skewness.