Question 1202686: In the country of United States of Heightlandia, the height measurements of ten-year-old children are approximately normally distributed with a mean of 56.7 inches, and standard deviation of 6.7 inches.
What is the probability that the height of a randomly chosen child is between 41.65 and 50.85 inches? Do not round until you get your your final answer, and then round to 3 decimal places.
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Answer: 0.180 (approximate)
Due to rounding error, a slightly different value may be the answer your teacher wants (eg: 0.179).
Explanation:
mu = 56.7 = population mean
sigma = 6.7 = population standard deviation
Compute the z score when x = 41.65
z = (x - mu)/sigma
z = (41.65 - 56.7)/6.7
z = -2.246269
z = -2.25
Do the same for x = 50.85
z = (x - mu)/sigma
z = (50.85 - 56.7)/6.7
z = -0.873134
z = -0.87
The task of computing P(41.65 < x < 50.85) is roughly equivalent to computing P(-2.25 < z < -0.87)
Now I'll use a Z table.
This kind of table should be found in the appendix section of your textbook.
If you don't have your textbook with you, then you can refer to a table such as this
https://www.ztable.net/
That page provides examples of how to read that table. Let me know if you have questions about it.
Use such a table to find that,
P(Z < -2.25) = 0.01222
and
P(Z < -0.87) = 0.19215
Subtract the two results to compute the area between those z scores.
P(a < Z < b) = P(Z < b) - P(Z < a)
P(-2.25 < Z < -0.87) = P(Z < -0.87) - P(Z < -2.25)
P(-2.25 < Z < -0.87) = 0.19215 - 0.01222
P(-2.25 < Z < -0.87) = 0.17993
That leads us to P(41.65 < x < 50.85) = 0.17993
Roughly 17.993% of the children have a height between 41.65 inches and 50.85 inches
0.17993 then rounds to 0.180 when rounding to three decimal places.
Depending on which table you use, or if you used a stats calculator, your final answer may be slightly different due to rounding error.
This link shown here
https://davidmlane.com/normal.html
offers a way to compute the answer fairly quickly. It also provides a nice diagram.
When I used that calculator, I got approximately 0.1789 which rounds to 0.179. That's not too far from the 0.180 I got earlier.
Here's an article talking about the TI84 and the normal cdf function
https://www.statology.org/normal-probabilities-ti-84-calculator/
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