Suppose that heights of newborns are bell shaped with a mean of 6.8 pounds and a standard deviation of 1.3 pounds. What percentage of weights are:
a) Below 6.8 lbs.?
b) Above 8.3 lbs.?
a) In a normal distribution, the mean = median = mode. What percentage are below the median?
Z = (score-mean(/SD
Calculate the Z score for b.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to your Z score.
To find the percentage of weights below or above certain values, we can use the standard normal distribution table or Z-table. The standard normal distribution table provides the cumulative probabilities associated with various z-scores.
In order to use the table, we need to convert the weights to z-scores. The formula to convert a value to a z-score is:
z = (x - μ) / σ
where:
- z is the z-score
- x is the value we want to convert
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
Now let's calculate the percentages:
a) To find the percentage of weights below 6.8 lbs, we need to find the z-score associated with the weight 6.8 lbs and then look up the cumulative probability in the standard normal distribution table.
z = (6.8 - 6.8) / 1.3
z = 0
The z-score of 0 corresponds to the mean of the distribution. Since we want the percentage below 6.8 lbs, we need to find the cumulative probability up to the z-score of 0. From the standard normal distribution table, the cumulative probability for a z-score of 0 is 0.5, or 50%.
Therefore, the percentage of weights below 6.8 lbs is 50%.
b) To find the percentage of weights above 8.3 lbs, we follow similar steps.
z = (8.3 - 6.8) / 1.3
z = 1.15
The z-score of 1.15 corresponds to a value above the mean. We want to find the percentage above 8.3 lbs, so we need to find the cumulative probability beyond the z-score of 1.15. From the standard normal distribution table, the cumulative probability for a z-score of 1.15 is approximately 0.8749, or 87.49%.
Therefore, the percentage of weights above 8.3 lbs is approximately 87.49%.