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STA 2023
CH. 2
• Find the mean, median, and standard
deviation of the data set { 5, 4, 7, 3, 1 }
Mean = 4
Median= 4
Std. dev.= 2.24
CH. 2
• If the mean is greater than the median, the
distribution is skewed right.
• If the mean is less than the median, the
distribution is skewed left.
• If the mean is equal to the median, the
distribution is not skewed.
CH. 2
• A football team scores an average of 20 points a game with a
standard deviation of 3 points. The distribution is
approximately normal. (a) What is the probability that the
team scores between 17 and 23 points? (b) Between 14 and
23 points?
a.) About 68%
b.) About 81.5%
CH.2
• The mean of a data set is 8 with a std. dev. of 1.5. If we
don’t know the shape of the distribution, what is the
probability between 6.5 and 9.5? Between 5 and 11?
• Chebychev: 1- 1/k2 Where k is the number of St. Devs
away from the mean.
Answer= At least 0%
Answer= At least 75%
CH. 3
• What is the difference between multiplicative,
combination, and permutation?
• Multiplicative: Just looking to find how many
possible outcomes per trial “x” from “n” number
of trials. (X)n
• Combination: Number of ways we can select n
items from N items without replacement. Use
𝑁!
when order doesn’t matter.
𝑛! 𝑁−𝑛 !
CH. 3
• Permutation: Number of ways we can put n
things out N in order. Use when order
matters!
•
𝑁!
𝑁−𝑛 !
CH. 3
• How many ways can we choose 5 flavors of ice
cream out of 21? How many ways can we put
3 flavors in order from first to third?
Answer:
𝑁!
𝑛! 𝑁−𝑛 !
Answer:
𝑁!
𝑁−𝑛 !
=
=
21!
5! 21−5 !
21!
21−3 !
= 20,349
= 7980
CH. 3
Currently Whipping Currently Nene
Total
Done Whipping
3
15
18
Done with Nene
7
8
15
Total
10
23
33
Find Probability of these dance combinations according to the
table:
(a) Done whipping or currently Nene;
(b) Finished Nene and currently Nene
CH. 3
• Answer (a):
0.79
A+𝐵− 𝐴∩𝐵
𝑇𝑜𝑡𝑎𝑙
=
• Answer (b):
𝐴∩𝐵
𝑇𝑜𝑡𝑎𝑙
= 0.24
=
8
33
18+23−15
33
=
26
33
=
CH. 3
• If P(A)= 0.3, P(B)= 0.5, and P(A U B)= 0.65:
(a) what is P(A ∩ B)?
(b) Are A and B independent?
𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 ∩ 𝐵)
CH. 3
a.) 𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃 𝐴 ∪ 𝐵 =
0.3 + 0.5 − 0.65 = 𝟎. 𝟏𝟓
b.) If independent, 𝑃 𝐴 𝑃 𝐵 = 𝑃 𝐴 ∩ 𝐵
0.3 0.5 = 0.15
Since 𝑃 𝐴 𝑃 𝐵 = 𝑃 𝐴 ∩ 𝐵 , A and B are
independent
CH. 4
X
2
3
5
7
P(x)
0.20
0.30
0.35
0.15
(a) Find μ and σ
(b) Find P(x=5); P(x≤ 7); P(x > 2)
𝜇=𝐸 𝑥 =∑ 𝑥 𝑃 𝑥
𝜎=
𝜎2 =
Σ 𝑥−𝜇
a.) μ= 4.1; σ= 1.67
b.) P(x=5)= 0.35; P(x≤ 7)= 1; P(x > 2)= 0.8
2
(𝑃 𝑥 )
CH. 4
You play roulette and place a $10 bet on red.
There are 18 red spaces, 18 black spaces, and 2 green spaces.
What is the expected result of your bet?
Probability of red (success)= 18/38
Probability of black (failure) = 18/38
Probability of green (failure) = 2/38
Success= 18/38
Failure= 20/38
CH. 4
(gain) x P(success) – (lose) x P(failure)
(10) X (18/38) – (10) x (20/38)
4.74 – 5.26 = -$0.52
We should expect to lose money.
CH. 4
• How do you know when to use binomial or
Poisson?
Binomial: 2 possible outcomes (success and fail).
𝑛!
𝑝 𝑥 𝑞 𝑛−𝑥
𝑛−𝑥 ! 𝑥 !
Poisson: Use when dealing with time/rate.
𝛾 𝑥 𝑒 −𝛾
𝑥!
where
𝛾 = 𝑚𝑒𝑎𝑛 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑒𝑠, and x=number of successes
we are interested in.
CH. 5
The average semester grade in STA 2023 is 70 with a
standard deviation of 3.5. What is the probability that the
average grade this semester will be greater than 76? Less
than 72? Equal to 68?
Answer: 0.50 – 0.4564 = 0.0436
Answer: 0.50 + 0.2157 = 0.7157
Answer: Cannot be equal to a number because these are
continuous random variables. P = 0.
CH. 5
The average grade in STA 2023 is 70 with a
standard deviation of 6. If you want to be in the
98th percentile, what is the minimum score you
must obtain?
Answer: 85.3
CH. 6
• We want to find out how many times Detroit
Lions fans cry themselves to sleep per week. We
randomly sample 40 Lions fans from a population
with a mean of 5 and std. dev. of 1.5. What is the
probability that the mean of our sample will be
more than 4.5? Less than 4?
P= 0.50 + 0.4826 = 0.9826
P= Our Z-score is -4.22. This is too negative to look
up, so we assume P<-4.22 = 0
CH. 7
• At a set level of confidence, does our
confidence interval increase or decrease as
sample size increases?
𝐶. 𝐼. = 𝑥 ± 𝑆. 𝐸.
𝑆. 𝐸. = 𝑍
𝜎
𝑛
Answer: Decreases
CH. 7
• At a set sample size, what happens to our
confidence interval as our level of confidence
increases?
𝐶. 𝐼. = 𝑥 ± 𝑆. 𝐸.
𝑆. 𝐸. = 𝑍
𝜎
𝑛
Answer: Increases
CH. 7
• If constructing interval for 𝜇:
When n> 30: 𝐶. 𝐼. = 𝑥 ± 𝑍𝛼/2
When n< 30: 𝐶. 𝐼. = 𝑥 ± 𝑡𝛼/2
• If finding n:
𝑍2𝜎 2
𝑛=
𝑆𝐸 2
𝜎
𝑛
𝜎
𝑛
CH. 7
• If constructing interval for P:
When n> 30: 𝐶. 𝐼. = 𝑝 ± 𝑍𝛼/2
When n< 30: 𝐶. 𝐼 = 𝑝 ± 𝑡𝛼/2
𝑝𝑞
𝑛
𝑝𝑞
𝑛
• If finding n:
𝑛=
𝑍𝛼/2
2
𝑝𝑞
𝑆𝐸 2
Note: When finding n, if 𝒑 is unknown use 0.50.
CH. 7
Construct a C.I. at 95% confidence with a sample
mean of 70 and a std. dev. of 20 when n=49.
When n=25
Answer = 70 ± 5.6 = (64.4, 75.6)
Answer = 70 ±2.064
20
25
= 70 ± 8.26
CH. 7
30 pre-med students out of a sample of 40 say they
have stress-induced acid reflux. Construct a 90%
confidence interval to estimate the true proportion
of pre-med students with stress-induced acid reflux.
𝑝 ± 𝑍𝛼/2
𝑝𝑞
𝑛
Answer = 0.75 ± 0.11 = (0.64, 0.86)
CH. 7
• Determine the sample size needed to
construct a 99% C.I. to estimate the true
proportion to within 0.10 with 𝑝 = 0.60. What
if we didn’t know 𝑝 ?
• 𝑆. 𝐸. = 𝑍𝛼/2
• 𝑛=
𝑍𝛼/2
2
𝑆.𝐸.2
𝑝𝑞
𝑝𝑞
𝑛
= 𝟏𝟔𝟎
CH. 8
It is believed that the average grade on STA 2023
final exams is 70. A study of 36 students was run,
and the results yielded a mean of 76 with a
standard deviation of 18. Is this enough evidence to
claim that the true mean score is greater than 70 at
𝛼 = 0.05? What is the level of significance?
Answer: Yes, our test statistic lies in the RR;
P=0.0228.
CH. 8
It is estimated that 70 percent of college
students enjoy going to Chipotle. A sample was
conducted where 23 out of 29 students sampled
said that they enjoy Chipotle. Is this enough
evidence to say that more than 70 percent like
Chipotle at 𝛼 = 0.1.
Answer: No, our T.S. of 1.09 does not fall in the
RR t>1.28.
CH. 9
A study was run to see if there is a difference in
mean test scores between students who play
piano, and students who do not. 20 piano
students and 18 non-piano students were
studied. The mean of the piano group was 85
with s.d. of 8, and the mean of the non-piano
group was 81 with s.d. equal to 7.5. Is this
enough evidence to conclude that there is a
difference at 𝛼 = 0.05? Construct a 95% C.I.
CH. 9
𝑠𝑝 2
𝑠1 2 𝑛1 − 1 + 𝑠2 2 𝑛2 − 1
=
𝑛1 + 𝑛2 − 2
T.S.=
𝑥1 −𝑥2
𝑠𝑝
2
1
1
+
𝑛1 𝑛2
Answer: Test stat is not in RR, do not reject.
Answer: 4 ± 5.23 = (−1.23,9.23)
Hopefully you now feel less like this
And more like Sheldon
• https://www.youtube.com/watch?v=ay3dSzkf
swE