the head circumferences of adult males have a bell-shaped distribution with the mean= 65cm and standard deviation =2cm.
a. what is the head circumference such that only 2.5% of adult males have a smaller head circumference?
b. what is the head circumference such that only 2.5% of adult males have a larger head circumference?
c. what is the head circumference such that only 16% of adult males have a larger head circumference?
To answer these questions, we can use the concept of standard deviation and the z-score. The z-score represents the number of standard deviations an observation or data point is from the mean. By finding the z-score corresponding to the given probabilities, we can then use it to calculate the corresponding head circumference.
a. To find the head circumference such that only 2.5% of adult males have a smaller head circumference, we need to find the z-score corresponding to the lower tail probability of 0.025. We can use the z-score formula:
z = (X - μ) / σ
Where:
z is the z-score,
X is the observed value (head circumference),
μ is the mean,
σ is the standard deviation.
Substituting the given values:
z = (X - 65) / 2
We need to find the value of X for which P(X < X_value) = 0.025. By looking up the z-score table or using a statistical calculator, we can find that the z-score corresponding to a lower tail probability of 0.025 is approximately -1.96.
-1.96 = (X - 65) / 2
Solving for X:
X - 65 = -1.96 * 2
X - 65 = -3.92
X ≈ 61.08 cm
Therefore, the head circumference such that only 2.5% of adult males have a smaller head circumference is approximately 61.08 cm.
b. To find the head circumference such that only 2.5% of adult males have a larger head circumference, we need to find the z-score corresponding to the upper tail probability of 0.025. Similarly, we can find that the z-score corresponding to an upper tail probability of 0.025 is approximately 1.96.
1.96 = (X - 65) / 2
Solving for X:
X - 65 = 1.96 * 2
X - 65 = 3.92
X ≈ 68.08 cm
Therefore, the head circumference such that only 2.5% of adult males have a larger head circumference is approximately 68.08 cm.
c. To find the head circumference such that only 16% of adult males have a larger head circumference, we need to find the z-score corresponding to the upper tail probability of 0.16. By looking up the z-score table or using a statistical calculator, we can find that the z-score corresponding to an upper tail probability of 0.16 is approximately 0.994.
0.994 = (X - 65) / 2
Solving for X:
X - 65 = 0.994 * 2
X - 65 = 1.988
X ≈ 66.988 cm
Therefore, the head circumference such that only 16% of adult males have a larger head circumference is approximately 66.988 cm.