Suppose a variable, Y , is normally distributed with a mean of 457.6and a standard deviation of 34.2. What percentage of values are between 425.6 and 489.6?
To solve this problem, we need to find the area under the normal curve between the values 425.6 and 489.6.
First, we calculate the z-scores for each value using the formula:
z = (x - μ) / σ
where x is the given value, μ is the mean, and σ is the standard deviation.
For 425.6:
z1 = (425.6 - 457.6) / 34.2 = -0.933
For 489.6:
z2 = (489.6 - 457.6) / 34.2 = 0.935
Using a standard normal distribution table or a calculator, we find the area to the left of z1 is approximately 0.1746 and the area to the left of z2 is approximately 0.8212.
To find the area between these two values, we subtract the smaller area from the larger area:
0.8212 - 0.1746 = 0.6466
Finally, we multiply this result by 100 to express it as a percentage:
0.6466 * 100 = 64.66
Therefore, approximately 64.66% of the values are between 425.6 and 489.6.