Ch. 8 Stats Vocab

inferences

assumptions made about a population based on your sample

• in order to generalize to the population, must be representative and random

point estimate

the statistic, the approximate value of the true proportion of the population

margin of error/standard error of statistic

measure of how the estimate may differ from the true population parameter

• indication of the precision of sample

• ± a value

calculating margin of error

(z*)(p̂q̂/n)^1/2

• z* = critical value —> C% of area is btwn ± z*

• p̂ = sample proportion

• q̂ = 1 - sample proportion

• n = number in sample size

confidence interval

estimate of a parameter, when reported as an interval of values

• gives an interval of plausible values for a parameter based on sample data

one-sample z interval

type of confidence interval —> point estimate ± margin of error

• statistic ± (critical value)(standard error of statistic)

• p̂ ± (z*)(p̂q̂/n)^1/2

  • z* = critical value —> C% of area is btwn ± z*

  • p̂ = sample proportion

  • q̂ = 1 - sample proportion

  • n = number in sample size

• note: if using calculator —> x = np̂, must round up to include decimal no matter what

conditions for one-sample z intervals

1. Random

2. large counts condition (np̂ ≥ 10, nq̂ ≥ 10) —> approximately normal

3. 10% Rule (n ≤ 10%N) —> ensures independence

how to find z*

use inverse normal to find area that is equivalent to the confidence level + ½ the extra

• ex: 95% confidence —> inverse normal of 95% + 5%/2 = z*

plausible

shouldn’t be surprised if the values in the interval are equal to the value of the parameter

• does NOT mean same thing as possible

confidence level C

the overall succes rate of the method used to calculate the confidence interval

• in C% of all possible samples, the interval computed from the sample data will capture the true parameter value

how to answer confidence interval problems

PANIC!

• P = parameter, define the parameter of interest

• A = assumptions & conditions, confirm all assumptions and conditions are fufilled

• N = name the interval type (ex: one-sample z-interval)

• I = interval calculation, calculate the interval by hand or with calculator

• C = context, write the problem in context

Note: Msut do for EVERY confidence interval question that requires constructing and interpreting a confidence interval

confidence interval in context

We are C% confident that the interval a to b captures the true proportion of (population of problem)

evaluating claims based on confidence intervals

If the upper and lower bounds are within the range of the claim —> agree

If either/both the upper and lower bounds are NOT within the claim —> disagree

• ex: majority —> lower bound and upper bound must be above 50%

  • if lower bound is below 50%, there is still plausible (unsurprising) value where the true propotion is below

what does increasing confidence level do?

increases the margin of error

• need to get larger for more possibility of capturing the population parameter

what does increasing sample size (n) do?

decreases the margin of error

• as sample size increases, variability of sampling distribution goes down

  • extreme values become offset by increasing frequency of non-extremes

confidence level in context

If many, many or a large # of confidence intervals were constructed (with the same method of sampling type, sample size, and interval type) for each, about C% of the confidence intervals would capture the parameter (use context)

how to choose sample size

1. use a proportion based on a previous study or similar study

2. use p̂ = 0.5 as a guess

   1. margin of error = largest when p̂ = 0.5 —> conservative guess

   2. ensures the margin of error is large enough, as it gets smaller with other proportions

calculating sample size for desired margin of error

(z*)(p̂q̂/n)^1/2 ≤ margin of error

• z* = critical value —> C% of area is btwn ± z*

• p̂ = sample proportion

• q̂ = 1 - sample proportion

• n = number in sample size

use algebra to solve for “n,” make sure to round UP