Chapter 5
What happens when a sample distribution is transformed into z scores?
-Shape does NOT change
-Mean turns to zero
-Standard deviation turns to one
For a normal distribution, what z-score separates the top 25% from the remainder of the distribution?
Z= .67
If a sample (n=50) was selected from a population with a mean of 63, the expected value of M in that distribution would be
63
Draw a distribution and label the critical region.
Tell me where you reject the null hypothesis and where you FTR the null hypothesis.
(use assumed alpha)
I have to see what you drew but basically your critical region (aka rejection region) is where you reject the null hypothesis.
What popular young adult book series sends “tributes” to participate in a televised competition in which they fight to the death?
The Hunger Games
A distribution of scores has a mean of μ=100 and a standard deviation of σ= 10. What z-score corresponds to a score of X=130 in this distribution?
z= 1.29
Random sampling vs random assignment
Sampling: Everybody in the population has
equal chance of being selected in the
sample
Assignment: Everyone in
population/sample has equal chance of
being assigned to groups
There is more variability in a distribution of ______ and less variability in a distribution of _______
Scores/ samples; Means
What are the Z-scores that correlate which each alpha level:
α=.05
α= .01
α= .001
.05 = -1.97, +1.96
.01 = -2.58, +2.58
.001= -3.30, +3.30
What color are Dr. V's eyes?
Green
For a distribution with a mean of μ=60 and σ=8, what value corresponds to a z-score of z=-1.50?
X= 48
On a statistics exam, the scores are normally distributed with a mean of μ=75 and a standard deviation of σ=8.
A student earns a score of X=83.
Report the students percentile rank.
Percentile Rank=84.13%
The law of large numbers states that
(Hint: size matters)
the larger the sample size (n), the more probable it is that the sample mean will be close to
the population mean
Larger discrepancies between sample mean and original population mean = higher or lower Z-scores
More variability = higher or lower standard error
Higher z-scores
Higher/larger standard error
Who won the 2022 World Cup?
Argentina
A score that is 9 points above the mean corresponds to a z-score of z=1.50. What is the population standard deviation?
σ=6
Describe independent random sampling.
Explain it in practice and in theory
(Hint: expectation vs reality)
Probability of being selected stays constant
from one selection to the next when more
than one individual is selected
*Requires sampling with replacement*
In theory, every individuals should have an equal chance but in practice, this isn't always the case
According to the Central Limit theorem, if you took __ or more samples from a population, what would happen to the shape of a negatively skewed distribution?
-The average value of all the sample means would be equal to __________
-_______ describes the variance between a sample mean and the true population mean
30; It would become/approach a normal distribution
-Population mean
-Standard error
Type 1 error happens when _____
Type 2 error happens when _____
_____ : the probability of type 1 error.
_____: the probability of type 2 error.
A researcher rejects the null hypothesis, when the null hypothesis is actually true.
A researcher fails to reject the null hypothesis, when the null hypothesis is false.
Alpha
Beta
What was the first Pixar movie?
Toy Story
For each of the following, identify which exam score should lead to the better grade.
a. A score of X=72 on an exam with μ=70 and σ=3
or
b. a score of X=78 on an exam with μ=70 and σ=12?
Both got the same grade.
Calculate probabilityThe highway department conducted a study measuring driving speeds on a local section of interstate highway. They found the average speed of μ=58 miles per hour with a standard deviation of σ=10. The distribution was approximately normal. Given the information, what proportion of the cars are traveling between 55 and 65 miles per hour?
.3759
There are 625 movie theaters in Colorado. The mean number of seats per theater is μ = 245.7. Suppose that the true population mean (μ = 245.7) and the standard deviation (σ = 52.4) are known to the Colorado film commission. They select a simple random sample of 40 movie theaters located within the state to estimate μ. The mean number of seats per theater in the sample is M = 252.9, with a sample standard deviation of s = 49.2.
a. The standard deviation of the distribution of sample means = __________
b. The standard or typical average difference between the mean number of seats in the 625 movie theaters in Colorado (μ = 245.7) and one randomly selected movie theater in Colorado is __________
c. The standard or typical average difference between the mean number of seats in the sample of 40 theaters (M = 252.9) and one randomly selected theater in that sample is __________
d. The standard or typical average difference between the mean number of seats in the 625 movie theaters in Colorado (μ = 245.7) and the sample mean of any sample of size 40 is __________
Population: N=625, μ=245.7, σ=52.4
Sample:n=40, M=252.9, s=49.2
a. 8.28
b. 52.4
c. 49.2
d. 8.28
Difference between population mean and sample mean= standard error
Difference between population mean and ONE theater= population standard deviation (sigma)
Difference between sample mean and ONE theater in that sample = sample standard deviation (s)
Although there is a popular belief that herbal remedies such as Ginkgo biloba and Ginseng may improve learning and memory in healthy adults, these effects are usually not supported by well controlled research. In a typical study, a researcher obtains a sample of n=16 participants and has each person take the herbal supplements every day for 90 days. At the end of the 90 days, each person takes a standardized memory test. For the general population, scores from the test form a normal distribution with a mean of 50 and a standard deviation of 12. The sample of research participants had an average of 54.
Assuming 2-tailed test, state HO and H1.
Conduct Z test and state whether the results were significant.
H0:μ=50
H1:μ≠50
Fail to reject the null hypothesis= not significant
X=54, μ=50, σ=12, n=16
Z= 1.33
French sculptor Frédéric-Auguste Bartholdi designed what U.S. landmark?
The Statue of Liberty