Answer:
a) 0.6406; b) 1.000; c) C. The probability from part (a) is more relevant because it shows the proportion of male passengers that will not need to bend; d) B. Since men are generally taller than women, a design that accommodates a suitable proportion of men will necessarily accommodate a greater proportion of women.
Step-by-step explanation:
For part a,
We use the formula for the z score of an individual,
Our mean, μ, is 69 and our standard deviation, σ, is 2.8. To find P(X < 70),
z = (70-69)/2.8 = 1/2.8 = 0.36
Using a z table, we see that the area under the curve to the left of this value is 0.6406. This is our probability.
For part b,
We use the formula for the z score of a sample,
Our mean is still 69 and our standard deviation is still 2.8. Our sample size, n, is 100. To find P(X < 70),
z = (70-69)/(2.8÷√100) = 1/(2.8÷10) = 1/0.28 = 3.57
Using a z table, we see the area under the curve to the left of this is greater than 0.9999. This means that the probability is 1.000.
For part c,
We want to know the proportion of male passengers that will not need to bend. This is why the value from part a is more important.
For part d,
Men taller on average than women. This means if we accommodate for the mean, it follows that the women will fit as well.