Answer:
"The probability that the egg will be with cholesterol content greater than 220 milligrams" is 0.37070 (37.070% or simply 37%)
Step-by-step explanation:
We have here a <em>normally distributed random variable</em>. The parameters that characterize this distribution is the <em>mean</em>, , and the <em>standard deviation</em>,
.
In this question, we have that:
milligrams.
milligrams.
And we want to know the probability that a randomly selected single egg "will be with cholesterol content greater than 220 milligrams."
To answer the latter, we need to use the following key concepts:
- Z-scores.
- The cumulative standard normal distribution, and
- The [cumulative] standard normal table.
The <em>z-scores</em> are standardized values and represent the distance (for the raw score) from the mean in standard deviations units. A <em>positive</em> z-score indicates that it is <em>above</em> and, conversely, a negative result that the value is <em>below</em> it.
The <em>cumulative distribution function</em> generates the values for the <em>cumulative standard normal distribution</em> displayed in the <em>standard normal table</em>.
The <em>standard normal distribution</em> is employed to find probabilities for any normally distributed data and we only need to calculate the corresponding z-score (or standardized value). This distribution has a and
.
As we can see, all of these concepts are intertwined, and each of them is important because:
- To find the corresponding probability, we first need to obtain the <em>z-score</em>.
- After this, we can consult the <em>standard normal table</em>, whose values are tabulated from the <em>cumulative standard normal distribution</em>, to find the requested probability.
Finding the probability
We can get the<em> z-score</em> using the formula:
[1]
Where <em>x</em> is the raw value we want to standardize using the previous formula, and, in this case is 220 milligrams, milligrams.
Thus (without using units):
To consult the <em>standard normal table</em>, we only need , because it only has values for two decimal digits. As a result, the value will be a little inexact (but near to the true value) compared to that obtained using statistical software (or maybe a more precise table).
With this value (which is positive and, therefore, above the mean), we need to carefully see the first column of the mentioned table to find z = 0.3. Then, in the first row, we only need to select that column for which we can add the next digit, in this case, 3 (it appears as +0.03). That is, we are finding the probability for .
Then, the <em>cumulative probability</em> for is:
However, the question is asking for "cholesterol content greater than 220 milligrams" or
Since
Which is the same for a standardized value:
Then
Therefore
Thus, "the probability that the egg will be with cholesterol content greater than 220 milligrams" is 0.37070 (37.070% or simply 37%).
The graph below shows a shaded area that corresponds to the found probability.
