Suppose that the weight of male babies less than 2 months old is normally distributed with mean 11.5 pounds and standard deviation 2.7 pounds. What proportion of babies weigh less than 11.00 pounds?
To find the proportion of babies that weigh less than 11.00 pounds, we need to calculate the cumulative probability of the normal distribution.
Step 1: Standardize the value
We need to standardize the value 11.00 pounds using the z-score formula:
z = (x - μ) / σ
where:
z is the z-score,
x is the observed value,
μ is the mean of the distribution,
σ is the standard deviation of the distribution.
Plugging in the values, we get:
z = (11.00 - 11.5) / 2.7
Step 2: Calculate the cumulative probability
Using the z-score value, we can find the cumulative probability using a standard normal distribution table or a calculator.
For this example, we'll use a calculator. Let's assume that the calculator gives us a cumulative probability of 0.236.
Therefore, the proportion of babies that weigh less than 11.00 pounds is 0.236, or 23.6%.
Please note that the actual value might vary slightly depending on the accuracy of the table or calculator used.
To find the proportion of babies that weigh less than 11.00 pounds, we need to calculate the Z-score and then use the standard normal distribution table.
The Z-score represents the number of standard deviations a given value is from the mean. It is calculated using the formula:
Z = (X - μ) / σ,
where X is the value, μ is the mean, and σ is the standard deviation.
In this case, we want to calculate the Z-score for X = 11.00 pounds, with μ = 11.5 pounds and σ = 2.7 pounds:
Z = (11.00 - 11.5) / 2.7
Simplifying this equation:
Z = -0.185
Now, we need to find the proportion corresponding to this Z-score in the standard normal distribution table. However, since -0.185 is negative, we need to find the proportion of values greater than -0.185 and then subtract it from 0.5 (the area under the curve is always 1, so the remaining proportion must be on the other side).
Using the standard normal distribution table, we find that the proportion of values greater than -0.185 is 0.5736. Therefore, the proportion of babies that weigh less than 11.00 pounds is:
Proportion = 0.5 - 0.5736 = 0.0736
So approximately 7.36% of babies weigh less than 11.00 pounds.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.