Answer:
a) 0.593; b) 0.453; c) 0.904; d) The sample size is larger which raises the probability.
Step-by-step explanation:
We find the z score for each of these problems. Each z score is for the mean of a sample rather than an individual value; this means we use the formula
The mean for each of these questions, μ, is 19; the standard deviation, σ, for each is 3.
For part a,
We want P(18.5 < X < 19.5). Our sample size, n, is 25. We find the z score of each endpoint, find the area under the curve to the left of each, and subtract them to find the area between the two values:
z = (18.5-19)/(3÷√25) = -0.5/(3÷5) = -0.5/0.6 = -0.83
z = (19.5-19)/(3÷√25) = 0.5/(3÷5) = 0.5/0.6 = 0.83
The area under the curve to the left of z = -0.83 is 0.2033, and the area under the curve to the left of z = 0.83 is 0.7967; this makes the area between them
0.7967-0.2033 = 0.5934 ≈ 0.593
For part b,
We want P(18 < X < 19). Our sample size is 25.
z = (18-19)/(3÷√25) = -1/(3÷5) = -1/0.6 = -1.67
z = (19-19)/(3÷√25) = 0/(3÷5) = 0/0.6 = 0
The area under the curve to the left of z = -1.67 is 0.0475, and the area under the curve to the left of z = 0 is 0.5000; this makes the area between them
0.5000 - 0.0475 = 0.4525 ≈ 0.453
For part c,
We want (18.5 < X < 19.5). Our sample size is 100.
z = (18.5-19)/(3÷√100) = -0.5/(3÷10) = -0.5/0.3 = -1.67
z = (19.5-19)/(3÷√100) = 0.5/(3÷10) = 0.5/0.3 = 1.67
The area under the curve to the left of z = -1.67 is 0.0475, and the area under the curve to the left of z = 1.67 is 0.9515; this makes the area between them
0.9515 - 0.0475 = 0.904
For part d,
The sample size in part c is 4 times larger than that of part a. This increases the probability.