Topic 2 : Confidence Intervals Flashcards
What is a confidence interval? (2)
2 components in 1 sentence
A type of inferential statistic that specifies a range of values that we use to estimate an unknown parameter
Point estimate+ex (2)
- A single statitic that is used as our best estimate of an unknown population parameter, based on sample
- E.g., mean number of Facebook friends among people under 35 is 220
Interval estimate + ex (2)
- A range of values within which the population parameter is likely to fall
- ex: people under 35 have between 200 and 240 Facebook friends
Most unbaised point estimate of the population mean μ is the
sample mean x̄
Level of Confidence
Specifies the probability that our interval estimate will in fact contain the population parameter
Confidence level +ex (2)
- Specifies the probability that the population parameter will indeed lie within that specfied range of values
- Ex: We are 95% confident that the true population mean number of facebook friends fall within a range of 200 to 240 friends
The higher the level of confidence, the — the confidence interval
wider
The higher the sample size, the —- the confidence interval
narrower
Sampling error (2)
What it is+ unknown
- The gap between sample statistics and the population parameters
- The true amount of sampling error is usually unkown because the parameter is unknown
The margin of error (E) is
an estimate of the maximum amount of difference that we think is possible between our statistic and its coresponding parameter
Difference between sampling error and margin of error
- sampling error is the gap betweeen sample statistics and population parameters while the MOE is an estimate of the maximium amount of difference we think is possible between our statistic and its coressponding parameter
critical value of Z for 90% confidence
1.645
What is C?
The area under the standard normal curve between critical values
The conditions for calculating a confidence interval for the mean:
- the distribution of the population is assumed to be normal
- Data is obtained by random sampling
Calculating margin of error when the standard deviation is known (3):
- Use Z and the normal distribution to determine the critical value
- use σ to calculate the standard error
- Multiple tgt
Calculating margin of error when the standard deviation is not known (3):
Calculation confidence intervals for μ (σ known) (4)
- Find the sample statistic (Sample mean)
- Find the critical Zc that coressponds to the given level of confidence
- Find the Margin of Error with [E= Zc (σ/√n)]
- Find the left and right end points and form the CI (x̄+/- E)
Explain the results of a 95% confidence interval
If a large number of samples is collected and a confidence interval is created for each sample, approximatelt 95% of these intervals will contain μ.
What if we raise the level of confidence? What is the trade off?
Your interval will get wider
What happens if you increase the sample size when calculating confidence interval? Given that standard deviation is the same? (2)
interval estimate+ Confidence interval
- The chances that your interval estimate does not contain the parameter will be reduced
- Confidence interval will be narrower
How to find the confidence interval if σ is unknown?
- When n> 30, can estimate σ with s and it will be reasonably accurate to use the Z-table to find the critical values and calculate our margin of error and CI
- When n < 30 or if we want to be more percise, we can use a distribution that has more area under the curve in the tails so we are not underestimating our margin of error
T- distribution (2)
What is it+ When is it used
- a family of curves each determined by a parameter called the degrees of freedom
- The t-distribution is used in statistics to estimate the significance of population parameters for small sample sizes or unknown variations
T distribution and sample size relationship
The T-distribution changes, depending on the sample size. The bigger the sample size, the more it looks like a normal distribution
As the degrees of freedom increases, the t-distribution
approaches the normal distribution
After —-, the t-distribution is very close to the standard normal Z-distribution
DF= 29 (n=30)
Tell me about the tails in the t-distribution and standard normal distribution
The tails in T-distributions are thicker to help us not underestimate our confidence interval
Steps for confidence interval for the population mean μ when σ unknown and n<30. (4)
- Find the margin of error using E= tc (S/√n)
- Find the left and right end points and form the CI (x̄+/- E)
Confidence intervals are a type of —-
inferential statistics
We use confidence interval for the —-
Distribu+organiz levels
- mean for normally distributed interval or ratio level variables
