Week 6 - Z scores III Flashcards
To calculate the probability of getting a value with a z-score between two other z-scores, you can either use a
reference table to look up the value for both scores and subtract them to find the difference, or you can use technology.
In this lesson, which is an extension of Z-scores and Z-scores II, we will practice both methods.
r
Historically, it has been very common to use a z-score probability table like the one below to look up the probability
associated with a given z-score:
analyze table in module
What is the probability associated with a z-score between 1.2 and 2.31?
To evaluate the probability of a value occurring within a given range, you need to find the probability of both the
upper and lower values in the range, and subtract to find the difference.
- First find z = 1.2 on the z-score probability reference above: .8849 Remember that value represents the
percentage of values below 1.2. - Next, find and record the value associated with z = 2.31: .9896
- Since approximately 88.49% of all values are below z = 1.2 and approximately 98.96% of all values are below
z = 2.31, there are 98.96% 88.49% = 10.47% of values between.
- What is the probability that a random selection will be between 8.45 and 10.25, if it is from a normal distribution
with µ = 10 and s = 2?
This question requires us to first find the z-scores for the value 8.45 and 10.25, then calculate the percentage of value
between them by using values from a z-score reference and finding the difference.
1. Find the z-score for 8.45, using the z-score formula: (xµ)
s
8.4510
2 = 1.55
2
⇡ 0.78
2. Find the z-score for 10.25 the same way:
10.2510
2 = 0.25
2
⇡ .13
3. Now find the percentages for each, using a reference (don’t forget we want the probability of values less than our
negative score and less than our positive score, so we can find the values between):
P(Z < 0.78) = .2177 or 21.77%
P(Z < .13) = .5517 or 55.17%
4. At this point, let’s sketch the graph to get an idea what we are looking for:
5. Finally, subtract the values to find the difference:
.5517.2177 = .3340 or about 33.4%
There is approximately a 33.4% probability that a value between 8.45 and 10.25 would result from a random
selection of a normal distribution with mean 10 and standard deviation 2.
Do z-score probabilities always need to be calculated as the chance of a value either above or below a given score?
How would you calculate the probability of a z-score between -0.08 and +1.92?
After this lesson, you should know without question that z-score probabilities do not need to assume only probabilities above or below a given value, the probability between values can also be calculated.
The probability of a z-score below -0.08 is 46.81%, and the probability of a z-score below 1.92 is 97.26%, so the
probability between them is 97.26% 46.81% = 50.45%.
What is the probability of a z-score between -0.93 and 2.11?
Using the z-score probability table above, we can see that the probability of a value below -0.93 is .1762, and the
probability of a value below 2.11 is .9826. Therefore, the probability of a value between them is .9826 .1762 =
.8064 or 80.64%
What is P(1.39 < Z < 2.03)?
Using the z-score probability table, we see that the probability of a value below z = 1.39 is .9177, and a value
below z = 2.03 is .9788. That means that the probability of a value between them is .9788.9177 = .0611 or 6.11%
What is P(2.11 < Z < 2.11)?
Using the online calculator on “Math Portal”, we select the top calculation with the associated radio button to the
left of it, enter “-2.11” in the first box, and “2.11” in the second box. Click “Compute” to get “.9652”, and convert
to a percentage. The probability of a z-score between -2.11 and +2.11 is about 96.52%.
What is the probability of a z-score between +1.99 and +2.02?
r
What is the probability of a z-score between -1.99 and +2.02?
r
. What is the probability of a z-score between -1.20 and -1.97?
r
. What is the probability of a z-score between +2.33 and-0.97?
r
What is the probability of a z-score greater than +0.09?
r
What is the probability of a z-score greater than -0.02?
r
What is P(1.42 < Z < 2.01)?
r
