sampling distribution
-repeat a random event (flip a coin) infinitely many times to generate a distribution
-shape of histogram is sampling distribution
-can talk about predicting the behavior of p?
-as you increase the number of repeats that you do (n), then p? becomes closer
sample proportion
p?
- p? is not necessarily p (it depends how many times you repeat the event)
- p?= # of observed successful outcomes/ # of total outcomes
- p? does not necessarily mean that landing on heads when flipping a coin = equal 50% (flip 2 times, not necessarily
population proportion
-true proportion
p
- p=50% for true proportion of heads when flip a coin
normal approximation to binomial distribution and sampling distribution
-normal approximation of binomial distribution is centered at mean (�) of np with a spread of ?(np(1-p))
-diff between sampling distribution and binomial probability is that sampling distribution talks about the proportion of success and not the number of
assumptions and conditions
-must check if these conditions are satisfied in order to use normal approximation (model) for the distribution of sample proportions
two assumptions (don't need to check but need to indicate you assume they're true)
1. independent assumption (the sample
The CDC reports that 22% of 18 y/o women in the US have a BMI of 25 or more- a value associated with an increased health risk. As part of a health check at a large college, the PE department usually requires students to come in to be measured and weighed.
p=0.22 (true proportion of females BMI of 25 or more)
p?=31/200=0.155 (how likely would it have been for me to seen this 15.5% if the true proportion was 22%?)
1. make a sampling distribution to answer this question where it is centered at the true p (0.2
Suppose that 13% of the population is left-handed. A 200-seat auditorium has been built with 15 "lefty-seats" seats that have the built-in desk on the left rather than the right arm of the chair.
In a class of 90 students, what's the probability that ther
p=0.13 = % of left-handed people
p?=15/90=0.167
1. make normal approximation to find the probability that there > 15/90 students will be left-handed
2. centered at 0.13 with a spread of ?(0.13)(0.87)/90 = 0.035
3. then find out when proportion of left han
mean vs. proportion
proportion is for categorical data (p?) and mean is for quantitative data (x bar)
1) suppose you have the following population {2,3,4,5,8,11}
2) suppose you create a sampling distribution for x bar (sampling mean) when you select a sample size of n=2 from this population (choose two random numbers from the data set) and record their sa
1) population mean (�) = 5.5
population st. deviation (?): 3.08 (how much #s deviate from the center)
population size (N): 6
2) randomly select 2 numbers out of the data set and record their average (all possible combinations for n=2 and their sample mean
unbiased estimator
average of all your x bars ends up being � (� of x bar = �)
-biased estimator if average of all your x bar does not equal � (� of x bar does not equal �)
-tells us why the center is �
the fundamental theorem of statistics
-the sampling distribution of any mean becomes more nearly Normal as the sample size grows
-if you have a large enough sample, your sampling distribution will be normal
-this is called the Central Limit Theorem
-not only does the histogram of the sample m
A college PE department asked a random sample of 200 female students to self-report their heights and weights, but the percentage of students with BMIs over 25 seemed suspiciously low. One possible explanation may be that the respondents 'shaded' their we
�=143.74
?=51.54
n=200 (which is greater than 30 so CLT guarantees that x bar is normal)
1. make normal distribution centered at 143.74 with a spread of ?/?n = 51.54/?200 = 3.64
2. you want to find the probability where you observe where x bar is less tha
The CDC reports that the mean weight of adult men in the US is 190 lbs, with a st. deviation of 59 lbs.
An elevator in our building has a weight limit of 10 persons or 2500 lbs. What's the probability that if 10 men get on the elevator, they will overload
(assume normal)
�=190
?=59
x bar: 2500/10 = 250 lbs (have to find for individual weight bc given � for individual weight)
1. probability that the average weight exceeds 250 (P(x bar ? 250))
2. normal distribution centered at 190 with a spread of 59/?10 =
categorical variable
parameter: p=population proportion
statistic: p hat = sample proportion
center: p=� of p hat
spread: ?pq/n
shape: normal if np ? 10 and nq ? 10
quantitative variable
parameter: �=population mean; ?=population st. deviation
statistic: x bar is sample mean
center: � = � of x bar
spread: ?/(?n)
shape: normal if n>30 (always normal if population is normal)