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Homework \# 4:
Question 11 of 40 (1 point) I Question Attempt: 1 of 3
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6
A normal population has mean and standard deviation .
(a) What proportion of the population is less than 20?
(b) What is the probability that a randomly chosen value will be greater than 6 ?
Round the answers to four decimal places.
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Part 1 of 2
The proportion of the population less than 20 is .
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Studdy Solution
STEP 1
What is this asking?
We're looking at a group of numbers (our population) with an average of 9 and a spread of 5, and we want to know how much of the group is less than 20 and the chance of picking a number bigger than 6.
Watch out!
Don't mix up the *less than* and *greater than*!
Also, remember to convert to z-scores before looking up probabilities.
STEP 2
1. Set up for less than 20
2. Calculate less than 20
3. Set up for greater than 6
4. Calculate greater than 6
STEP 3
We're given that the population **mean** is and the **standard deviation** is .
We want to find the proportion of the population less than .
This means we want to find .
STEP 4
To do this, we'll convert our value of to a **z-score** using the formula: This formula tells us how far away is from the mean, in terms of standard deviations.
STEP 5
**Plug in** our values , , and into the z-score formula: So, 20 is **2.2 standard deviations** above the mean.
STEP 6
Now, we look up in a **z-table** (or use a calculator) to find the probability .
This gives us the proportion of the population less than 20.
A z-table tells us the area to the *left* of the z-score, which is exactly what we want!
We find .
STEP 7
Now we want to find the probability that a randomly chosen value is greater than , which is .
We'll use the same strategy: convert to a z-score.
STEP 8
**Plug in** the values , , and into the z-score formula: So, 6 is **0.6 standard deviations** *below* the mean (notice the negative sign!).
STEP 9
Looking up in the z-table gives us .
However, we want , the area to the *right* of the z-score.
Since the total area under the curve is 1, we can find this by subtracting from 1:
STEP 10
(a) The proportion of the population less than 20 is approximately **0.9861**. (b) The probability that a randomly chosen value will be greater than 6 is approximately **0.7257**.
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