a. The probability that on a randomly selected day, the jail population
is greater than 750,000 is 20.1%
b. The probability that on a randomly selected day, the jail population is
between 600,000 and 700,000 is 43.2%
Step-by-step explanation:
The given is:
1. The average daily jail population in the United States is 706,242
2. The distribution is normal and the standard deviation is 52,145
3. We need to find the probability that on a randomly selected day,
the jail population is greater than 750,000
4. We need to find the probability that on a randomly selected day,
the jail population is between 600,000 and 700,000
a.
At first find z-score
∵ z = (x - μ)/σ, where x is the score, μ is the mean and σ is the standard
deviation
∵ x = 750,000 , μ = 706,242 and σ = 52,145
∴ z = [tex]\frac{750,000-706,242}{52,145}[/tex] ≅ 0.84
Use the normal distribution table of z to find the area to the right of
the z-value
∵ The corresponding area to z-score of 0.84 is 0.79955
- But we are interested in x > 750,000, we need the area to the
right of z-score
∴ P(x > 750,000) = 1 - 0.79955 = 0.2005
∴ P(x > 750,000) = 0.2005 × 100% = 20.1%
The probability that on a randomly selected day, the jail population is
greater than 750,000 is 20.1%
b.
We will find z-score for 600,000 < x < 700,000
∵ z = [tex]\frac{600,000-706,242}{52,145}[/tex] ≅ -2.04
∵ z = [tex]\frac{700,000-706,242}{52,145}[/tex] ≅ -0.12
Use the normal distribution table of z to find the area between
the two z-values
∵ The corresponding area to z-score of -2.04 is 0.02068
∵ The corresponding area to z-score of -0.12 is 0.45224
- To find P(600,000 < x < 700,000) subtract the two values above
∴ P(600,000 < x < 700,000) = 0.45224 - 0.02068 = 0.4316
∴ P(600,000 < x < 700,000) = 0.4316 × 100% = 43.2%
The probability that on a randomly selected day, the jail population is
between 600,000 and 700,000 is 43.2%
Learn more:
You can learn more about z-score in brainly.com/question/7207785
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There is a probability of 20% that the jail population is greater than 750,000
There is a probability of 43.15% that the jail population is between 600,000 and 700,000
The z score is used to determine by how many standard deviations the raw score is above or below the mean. The z score is given by:
z = (raw score - mean) / standard deviation
Given that mean = 706242, standard deviation = 52,145
a) For x > 750000
z = (750000 - 706242)/52145 = 0.84
P(z > 0.84) = 1 - P(z < 0.84) = 1 - 0.7995 = 0.2005
b) For x > 600000
z = (600000 - 706242)/52145 = -2.04
For x < 700000
z = (700000 - 706242)/52145 = -0.12
P(-2.04 < z < -0.12) = P(z < -0.12) - P(z < -2.04) = 0.4522 - 0.0207 = 0.4315
There is a probability of 20% that the jail population is greater than 750,000
There is a probability of 43.15% that the jail population is between 600,000 and 700,000
Find out more on z score at: https://brainly.com/question/25638875
