Suppose the length of time cats sleep each day is normally distributed with a mean of 15 hours and a standard deviation of 2 hours.
Which groups describe 34% of the cat population? I just need help on how to do it.
You must have been given some choices in the range to give you .34
To determine which groups describe 34% of the cat population, we can use the concept of the standard normal distribution.
1. Convert the given normal distribution to a standard normal distribution by using the formula z = (x - μ) / σ, where z is the z-score, x is the value, μ is the mean, and σ is the standard deviation.
2. In this case, we need to find the z-scores that correspond to 34% of the cat population. To do this, we can use a z-table (also known as a standard normal table) or a statistical calculator.
3. Since the normal distribution is symmetric around the mean, we can find the z-scores that correspond to 34% in both tails of the distribution.
4. Look for the z-scores in the z-table by finding the closest probability value to 0.34. The z-scores will correspond to the closest probability values found in the table.
5. Once you have the z-scores, use the formula z = (x - μ) / σ to find the values corresponding to those z-scores in terms of hours of sleep.
For example, to find the lower z-score (tail on the left side), look for the probability closest to 0.34 in the z-table. Let's say it corresponds to a z-score of -0.44.
Substituting this z-score into the formula and solving for x:
-0.44 = (x - 15) / 2
Solving for x, we get:
x - 15 = -0.44 * 2
x - 15 = -0.88
x = 15 - 0.88
x ≈ 14.12
Therefore, approximately 34% of the cat population sleeps less than 14.12 hours per day.
To find the upper z-score (tail on the right side), subtract the lower z-score that we found from 1 to get the remaining proportion.
1 - 0.34 = 0.66
Look for the closest probability of 0.66 in the z-table. Let's say it corresponds to a z-score of 0.41.
Substituting this z-score into the formula and solving for x:
0.41 = (x - 15) / 2
Solving for x, we get:
x - 15 = 0.41 * 2
x - 15 = 0.82
x = 15 + 0.82
x ≈ 15.82
Therefore, approximately 34% of the cat population sleeps more than 15.82 hours per day.
So, the two groups that describe approximately 34% of the cat population are those cats that sleep less than 14.12 hours per day and those that sleep more than 15.82 hours per day.