Calculating Densities

In the box below, please enter the value of F(22.692), i.e. of P[X ≤ 22.692]. We are given that X ~ Normal(μ=31; σ=4). When you have entered your value, click on the “Check your answer!” button. Please round your answer to the ten-thousandths place.

Your guess:

You got the correct answer of F(22.692) = 0.0189. Congratulations!

Unfortunately, your answer was not correct. Either try again or click on “Show Solution” below to see how to obtain the correct answer.

Important: Do not forget that F(x) is a probability. As such, its value must be between 0 and 1, inclusive. Your answer was outside this range. This is quite problematic.

Show Formula

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For the Normal distribution, the formula to calculate cumulative probabilities does not exist. The process is either to use technology or to perform a z-transform and use a Z-Table. Here, we do the latter. At the bottom of the page, you will find the code to perform these calculations using technology.

For the Normal distribution, the formula to calculate probability densities is

In this formula, there are several symbols to know:

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That part was always. This is particular to this problem:

Now that we know the value of \( z \), we go to our Z-Table to find the probability. Usually, one rounds the z-value to the closest hundredths. Find the z-value along the margins. The units and tenths values will be along the left side (-2.0), the hundredths value will be along the top (0.08). The cumulative probability is the number at the crossing of the row and column. Here, that value is 0.018906.

And so, from this we know that the probability of the next piece of candy weighing no more than 22.692 grams is approximately 1.8906%.

Show the R Code

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Copy and paste the following code into your R script window, then run it from there.

In the R output, the number given after running the final line above is the probability density requested, F(22.692). There are three slots in the function. The first is x; the second, μ; the third, σ. Having R perform the calculation increases the precision (and accuracy) of the calculation.

Show the Excel Formula

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Copy and paste the following code into an Excel cell, then hit enter.

The first slot is the value of x. The second slot is the value of μ. The third slot is the value of σ. The last slot is TRUE for calculating F(x), P[X ≤ x].