In a cross country meet, you run at a steady pace of 7 minutes a mile. If you finish the race in 21 minutes. How long is the rac
2 answers:
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Consider the system of inequalities
1. Plot all lines that are determined by equalities (see attached diagram)
2. Determine which bounded part of the plane you should select:
means that you should take points with y-coordinates greater than or equal to 2 (top part of the coordinate plane that was formed by the red line);
means that you should take points with x-coordinates less than or equal to 6 (left part of the coordinate plane that was formed by the blue line);
- for
you can check where the origin is placed. Since
, the origin belongs to the needed part and you have to take the right part of the coordinate plane that was formed by green line.
- for
you can check where the origin is placed. Since
, the origin belongs to the needed part and you have to take the bottom part of the coordinate plane that was formed by orange line.
3. According to the previous explanations, the shaded region is as in A diagram.
Answer: correct choice is A.
Answer:
a) P(7.99 ≤ X ≤ 8.01) = 0.8164
b) P(X ≥ 8.01) = 0.0475.
Step-by-step explanation:
To solve this question, we have to understand the normal probability distribution and the central limit theorem.
Normal probability distribution:
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean and standard deviation
, the zscore of a measure X is given by:
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central limit theorem:
The Central Limit Theorem estabilishes that, for a random variable X, with mean and standard deviation
, the sample means with size n can be approximated to a normal distribution with mean
In this problem, we have that:
(a) Calculate P(7.99 ≤ X ≤ 8.01) when n = 16.
n = 16, so
This probability is the pvalue of Z when X = 8.01 subtracted by the pvalue of Z when X = 7.99. So
X = 8.01
Applying the Central Limit Theorem
has a pvalue of 0.9082
X = 7.99
has a pvalue of 0.0918
0.9082 - 0.0918 = 0.8164
P(7.99 ≤ X ≤ 8.01) = 0.8164
(b) How likely is it that the sample mean diameter exceeds 8.01 when n = 25? P(X ≥ 8.01) =
n = 25, so
This is 1 subtracted by the pvalue of Z when X = 8.01. So
has a pvalue of 0.9525
1 - 0.9525 = 0.0475
P(X ≥ 8.01) = 0.0475.