A
Provide an appropriate response.
Which of the following cannot be the probability of an event?
(a) 0 (b) 0.001 (c) - 0.35 (d) 1
(c) - 0.35
Do the followings represent a probability distribution?
x P(x)
0 0.0625
1 0.2500
2 0.3750
3 0.2500
4 0.0625
1. The sum of all probabilities must be 1 “∑ 𝑃(𝑥) = 1" 𝑤ℎ𝑒𝑟𝑒 𝑥 𝑎𝑠𝑠𝑢𝑚𝑒𝑠 𝑎𝑙𝑙 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑣𝑎𝑙𝑢𝑒𝑠
2. Each probability value must be between 0 and 1 inclusive 0 ≤ 𝑃(𝑥) ≤ 1 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑎𝑛𝑑𝑜𝑚 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑥
So, yes!
Using Standard Normal distribution table, find the indicated probabilities with the given Z scores.
1. Less than -2.75
2. Greater than 1.96
3. Between 2.00 and 2.13
HINT: What should we use on our calculator? How do we get there?
1. normalcdf (-999, -2.75, 0, 1)= 0.0030
2. normalcdf (1.96, 999, 0, 1)= 0.0250
3. normalcdf (2.00, 2.13, 0, 1)= 0.0062
What are the three freebies of Confidence Level
90% , 95% , 99%
Find critical Z values. Assume normal distribution.
a) Two tailed test, a=0.01
HINT: Your first step should be to graph.
invNorm
Left side of area, mean, sd
Two tailed, 0.01/2
invNorm (.005, 0, 1)= ± 2.575
The following frequency distribution analyzes the scores on a math test. Find the class boundaries from all scores.
39.5, 59.5, 75.5, 82.5, 94.5, 99.5
What are the steps to find the Mean and SD from the Probability Distribution on your calculator?
Enter the data [L1: x & L2: P (x)] → CALC → 1: 1-Var Stats → Enter the List TI-83: L1, L2 or TI-84: List (L1) and FreqList (L2)
SAT Math scores roughly follow a normal distribution with mean of 505 and a standard deviation of 110. How high must a student score to be placed in the top 10% of all students?
HINT: Sketch out a normal distribution curve!
HINT: Should we do this manually or the calculator way? Where do I go on my calculator?
invNorm (1-0.1, 505, 101)
x= 645.97
Score of 646
BONUS: If we do this problem manually, what do we need?
A genetic experiment with peas resulted in one sample of offspring that consisted of 438 green peas and 163 yellow peas.
Construct a 90% confidence interval to estimate the percentage of yellow peas.
FIND THE KEY WORD, WHAT CALCULATOR STEP TO USE, AND LASTLY WHAT'S THE ANSWER!
Step 1) STAT - TEST - A:1-PropZInt
0.241 < p <0.301
You spin a spinner. Find the probability spinning an odd number or even number.
HINT: THE SPINNER HAS THE NUMBERS 1-9
P(ODD OR EVEN)= 5/9+4/9=1
BONUS: What does this tell you about your chances of spinning an odd or even number?
The following frequency distribution analyzes the scores on a pulse rates. Construct a relative percentage table.
6%, 54%, 28%, 8%, 2%, 2%
What are the steps to find Binom probability on your calculator?
DISTR (2nd VARS) → 0: binompdf (or A: binomcdf) → Enter the values for “n, p, x” or “binompdf (n, p, x)”
A certain airline currently has a seat width of 17 inches. Men have hip breadths that are normally distributed with a mean of 14.4 inch and a standard deviation of 1.0 inches.
Find the probability that if a man is randomly selected, his hips will be greater than 17 inches.
HINT: Sketch it out!
normalcdf (17, 999, 1.4, 1)= 0.00466
.47%
Archeologists have studied sizes of Egyptian skulls in an attempt to determine whether breeding occurred between different cultures. Listed below are the widths (mm) of skulls from 150 A.D. Construct a 95% confidence interval estimate of the mean skull width.
128.3, 138.1, 125.9, 132.1, 143.1, 135.3, 138.9, 128.7
FIND THE KEY WORD, WHAT CALCULATOR STEP TO USE, AND LASTLY WHAT'S THE ANSWER!
Step 1) STAT - EDIT - L1 (put the values)
Step 2) STAT - CALC - 1: 1-VAR STATS (for mean and SD)
Step 3) STAT - TEST - 8: TInterval
79.4 min < μ < 105.4 min
Find P-value.
The claim is that for Verizon data speeds at airports, the mean is μ=14.00 mbps. n=13 and the test statistic t=−1.625
HINT: What tail is the claim?
df=12 (df= n-1)
P-value= tcdf (-999, -1.625, 12) X2
= 0.130
Express 1 as a percent.
a)1% b) 10% c) 11% d) 100%
d) 100%
Based on a Harris poll, 60% of adults believe in the devil. Assuming that we randomly select five adults. Use the formula and a technology to find the following:
The probability that the number of adults who believe in the devil is at least three.
𝑃(𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝑡ℎ𝑟𝑒𝑒) = 𝑃(𝑥 ≥ 3) = 1 − 𝑏𝑖𝑛𝑜𝑚𝒄𝑑𝑓(5, 0.6, 2) = 𝟎. 𝟔𝟖3
A certain airline currently has a seat width of 17 inches. Men have hip breadths that are normally distributed with a mean of 14.4 inch and a standard deviation of 1.0 inches.
Another airline uses an aircraft that seats 122 passengers. If the plane is full with 122 random men, find the probability that these have a mean hip breadth greater than 17 inches.
HINT: Sketch it out!
Z score= ̅x - ℳ / (σ √ n)= 17-14.4/(1√ 122)= 28.7
normalcdf (17, 28.7, 14.4, 1/√ 122)= 0
a) Determine the critical value Za/2 corresponding to the 88% confidence level.
b) Determine the critical value for a left-tailed test of a population mean at the α = 0.05 level of significance based on a sample size of n = 35.
a) 1.555
b) -1.691
Use the given info to find the P-value.
The test statistic in a left tailed test is z=-1.72
Sketch it out!
normalcdf (-999, -1.72, 0, 1)= .0427
A flashlight has 6 batteries, 2 of which are defective. If 2 are selected at random without replacement, find the probability that both are defective.
HINT: What is the keyword?
P(both defective) 2/6 x 1/5= 2/30 (1/15) === .067
The brand name of McDonald’s has a 95% recognition rate (based on data from Retail Marketing Group and Harris Interactive). A special focus group consists of 12 randomly selected adults to be used for extensive market testing. For such random group of 12 people, consider recognizing the brand name as a success and answer the following:
Use the range rule of thumb to find the minimum usual number and the maximum usual number of people who recognize the brand name of McDonal’s.
𝑚𝑖𝑛: 𝜇 − 2𝜎 = 11.4 − 2(0.8) = 𝟗. 𝟖 𝒑𝒆𝒐𝒑𝒍𝒆
𝑚𝑎𝑥: 𝜇 + 2𝜎 = 11.4 + 2(0.8) = 𝟏𝟑 𝒑𝒆𝒐𝒑le
a) Find the mean 𝑛 = 12, 𝑝 = 0.95 𝜇 = 𝑛𝑝 = 12(0.95) = 𝟏𝟏. 𝟒 𝒑𝒆𝒐𝒑𝒍𝒆
b) Find the variance. 𝜎 2 = 𝑛𝑝𝑞 = (12)(0.95)(0.05) = 0.57 = 𝟎. 𝟔 𝒑𝒆𝒐𝒑𝒍𝒆 𝟐
c) Find the standard deviation 𝜎 = √𝑛𝑝𝑞 = √0.57 ≈ 0.75498 = 𝟎. 𝟖 𝒑𝒆𝒐𝒑𝒍𝒆
d) Find the expected value. 𝐸 = 𝜇 = 𝑛𝑝 = 12(0.95) = 𝟏𝟏. 𝟒 𝒑𝒆𝒐𝒑𝒍e
Round to one decimal place: 999.929875
999.9
Name the things that you can find using your calculator and things that you can't find directly using calculator (need to use formula)
With Calculator
~ Z-score (Critical Value) {DO NOT WASTE YOUR TIME FINDING Z-Score for THE FREEBIES}
~ Confidence Interval (both proportion & mean)
~ Point of estimation (both proportion & mean)
Without Calculator
~ Margin of Error (both proportion & mean)
~ Sample Size (both proportion & mean)
~ Point of estimation and margin of error when CI is given (both proportion & mean)
Harper’s Index reported that 80% of all supermarket prices end in the digit 9 or 5. Supposed you check a random sample of 115 items in a supermarket and find that 88 have prices that end in 9 or 5. Does this indicate that less than 80% of the prices in the store end in the digits 9 or 5? Use a 0.05 significance level.
1) State the null and alternative hypothesis
2) Find the critical value(s).
1) h0: p=0.8 h1: p<0.8
2) invNorm (0.05, 0, 1)
Zscore= -1.645