Sampling Distribution Basics
On Earth, approximately 11% of all 8 billion people are lefties. In a SRS of 400 people on Earth, 10.8% of people were lefties.
What is the parameter and what is the statistic in the above statement?
parameter: 11% of all people are lefties
statistic: 10.8% of those sampled were lefties
What symbol do we use for a sample proportion?
p-hat
You come across the following problem on a test:
"A machine that fills cereal boxes fills the boxes with a mean of 12 ounces with a standard deviation of 0.4 ounces. A random sample of 50 boxes has a mean of 11.6 ounces, determine if this is convincing evidence the machine is malfunctioning"
Explain why it is apparent this is a means problem
Any of the following reasons work:
1) we are told a standard deviation
2) Mass/number of ounces is a quantitative variable
3) we are discussing the mean from a sample, not the proportion with an attribute
The lifespan of female Ohioans is normally distributed with a mean of 81 and the st dev of 5.
For a SRS of 12 female Ohioans, explain why the sampling distribution for this would be approximately normal.
What is... the population distribution is known to be approximately normal
Who is headlining the Superbowl halftime show this weekend?
Who is Bad Bunny
On Earth, approximately 11% of all 8 billion people are lefties. In a SRS of 400 people on Earth, 10.8% of people were lefties.
What is the population and what is the sample in the above statement?
Population: all people on Earth
Sample: SRS of 400 people on Earth
You come across the following question on a test:
"It is known that 95% of bees in a city are female. In a simple random sample of 500 bees in the city, 435 are female. Determine if this is a surprising result"
Explain what in the problem tells you it is a PROPORTIONS problem instead of a MEANS problem
Any one of the following would work:
1) We are talking about the percent of a population that has a specific attribute, i.e. the proportion of the population that are female
2) We aren't told any standard deviation
3) this is a Binomial/CATEGORICAL problem: the success=female, fail=male
A machine that fills cereal boxes fills the boxes to a mean of 12 ounces with a standard deviation of 0.4 ounces.
If you were to create the sampling distribution for a random sample of 50 boxes, what would the mean and standard deviation of the sampling distribution be?
mean=12 oz
st dev= 0.4/sqrt(50)=0.057
It is known that 95% of bees are female. In a simple random sample of 50 bees in a hive, 43 are female.
Explain why WE CANNOT use the normal distribution to solve a probability problem for this scenario
We cannot use the normal distribution because this fails the large counts check....
0.95*50=47.5>10,
but 0.05*50=2.5 which is not at least 10
Complete the phrase:
"It's not clocking to you that I'm standing on ..."
"Business"
-Justin Bieber
What is a sampling distribution?
The distribution of the statistic of interest for every possible sample of a given size from a population
It is known that 95% of bees are female. In a simple random sample of 50 bees in a hive, 43 are female. Assume we sampled less than 10% of the hive.
Calculate the standard deviation for the sampling distribution of proportion of bees that are female, and interpret it.
What is 0.03
"In many many samples of 50 bees in a hive, the proportion that are female typically varies from the mean of 95% by about 3%" ?
Note: we have to be assured we sampled less than 10% of the hive to solve this problem!
What is the difference between x-bar, μx-bar, and μ ?
x-bar is the average of a sample
μx-bar is the average of a sampling distribution
μ is a population average
The distribution of wealth of Americans is strongly skewed right, with most people being worth a little and a few owning a large amount of wealth. Explain the shape of a sampling distribution of the average wealth of a SRS of 40 Americans
By the central limit theorem, since we sampled over 30 Americans, this distribution is approximately Normal
Approximately how many hours was TikTok banned/taken down in early 2025?
What is 12 hours
Increasing sample size does what to the sampling distribution?
Decreases variability
It is known that 8% of males are colorblind. For the sampling distribution for 200 randomly selected males, determine the mean and standard deviation.
mean=0.08
standard deviation=sqrt((0.08*0.93)/200)=0.192
In a population, the average age at which a child loses their first tooth is 7.3 year with a standard deviation of 0.5 years. You take a SRS of 12 children in the community and find the average age in your sample to be 6.8 years. Explain why you can't find the probability of this outcome.
We don't know the population is normally distributed, and we didn't sample at least 30 children, so we can't use the Normaldist.
Would you use the 10/10 large counts check, or the CLT for the following scenario to confirm the sampling distribution is approximately normal:
"In a school, approximately 67% of students graduate buy lunch each day. In a SRS of 100 students, what is the probability that over 70 bought their lunch on a given day?"
This is a proportions problem, you must use the 10/10 large counts check.
0.67*100>10 and 0.33*100>10
How many bones does a shark have?
What does it mean to be an unbiased estimator?
The mean of the sampling distribution is equal to the population parameter of interest
According to known data, 45% of all students like block schedules. A sample of 200 students is taken. What is the probability the sample proportion is between 50% and 60%? Assume all conditions for normality and independence are met.
Normaldist(mean=0.45, stdev=sqrt((.45*.55)/200) )
P(0.5<X<0.6)=0.078
Duracell batteries have an average charge of 98 hours and a standard deviation is 8. You sample 40 batteries and find the average charge in the sample to be 97. What's the probability of this average or worse? Is it surprising?
Normaldist(mean=98, st dev=8/sqrt(40))
P(X<97)=0.215
This is not a surprising result because this happens more than 5% of the time.
As of a December 2025 poll of 1000 likely Ohio voters, Amy Acton is edging out Vivek Ramaswamy with 46% of the votes (9% undecided, 45% Ramaswamy).
Explain why the sampling distribution of the proportion who plan to vote for Acton in this type of poll is approximately normally distributed.
First, it is independent because we sampled less that 10% of likely Ohio voters.
It is approximately normally distributed since 0.46*1000>10 and 0.54*1000>10
What Mediterranean dip is made from charred eggplant mixed with tahini, olive oil, lemon juice, and garlic
What is baba ganoush?