Answers:
These values correspond to problems 1 through 4, in order from top to bottom.
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Work Shown for Problem 1
mu = 92.12 = mean
sigma = 2.43 = standard deviation
z = (x - mu)/sigma
z = (85.4 - 92.12)/(2.43)
z = -2.76543209876542
z = -2.77
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Work Shown for Problem 2
On your TI84 calculator, press the button labeled "2ND" in the top left corner. Then press the button labeled "VARS". This button is next to the arrow keys. It's one spot to the left of "Clear".
Pressing "2ND" then "VARS" will lead to the stats distribution menu. Scroll down to "normalcdf" and press enter.
The template for this function is
normalcdf(lower, upper, mu, sigma)
We'll have these inputs
- lower = 88.7
- upper = 96.3
- mu = 92.12
- sigma = 2.43
This is what you should have typed into your calculator
normalcdf(88.7, 96.3, 92.12, 2.43)
Computing that will result in the approximate value of 0.8776454157
That rounds to 0.878 when rounding to 3 decimal places.
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Work Shown for Problem 3
The normalcdf function will be used again.
This time the inputs are:
- lower = -9999
- upper = 88.7
- mu = 92.12
- sigma = 2.43
The -9999 represents negative infinity. Replace it with any large negative value.
This is what you should have typed into your calculator
normalcdf(-9999, 88.7, 92.12, 2.43)
It results to 0.0796533584 and leads to 0.080 when rounding to 3 decimal places.
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Work Shown for Problem 4
The fastest 12% of the pitches pairs with the slowest 88%
We need to find the speed that corresponds to the 88th percentile, meaning we need to find the value of k such that P(x < k) = 0.88
We won't be using normalcdf, but we'll use the inverse function of this idea.
Press the button labeled "2ND" then "VARS" to get to the stats menu.
Scroll down to "invNorm"
The template for this function is
invNorm(area, mu, sigma)
In this case we have these inputs
- area = 0.88
- mu = 92.12
- sigma = 2.43
Therefore you should have this typed in:
invNorm(0.88, 92.12, 2.43)
The approximate result is: 94.9752179
It means P(x < 94.9752179) = 0.88 approximately
And also P(x > 94.9752179) = 0.12 approximately
Rounding 94.9752179 to the nearest tenth yields 95.0
Roughly 12% of the pitches are faster than 95 mph (approximate).
Let me know if you have any questions on how to use the TI84.