Respuesta :
Answer:
a. Their average weight is of 200 pounds.
b. 0.0875 = 8.75% probability that the maximum safe weight will be exceeded
c. 0.0002 = 0.02% probability that the maximum safe weight will be exceeded
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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 p-value, 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 normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
a. If 60 people are riding the coaster, and their total weight is 12,000 pounds, what is their average weight?
12000/60 = 200 pounds
Their average weight is of 200 pounds.
b. If a random sample of 60 adult men ride the coaster, what is the probability that the maximum safe weight will be exceeded?
Men means that [tex]\mu = 189, \sigma = 63[/tex].
Sample of 60 means that [tex]n = 60, s = \frac{63}{\sqrt{60}} = 8.13[/tex]
Safe weight will be exceeded if the sample mean is above 200 pounds, which has probability of 1 subtracted by the pvalue of Z when X = 200. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{200 - 189}{8.13}[/tex]
[tex]Z = 1.35[/tex]
[tex]Z = 1.35[/tex] has a pvalue of 0.9115
1 - 0.9115 = 0.0875
0.0875 = 8.75% probability that the maximum safe weight will be exceeded.
c. If a random sample of 60 adult women ride the coaster, what is the probability that the maximum safe weight will be exceeded?
Women means that [tex]\mu = 165, \sigma = 76[/tex]
Sample of 60 means that [tex]n = 60, s = \frac{76}{\sqrt{60}} = 9.81[/tex]
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{200 - 165}{9.81}[/tex]
[tex]Z = 3.57[/tex]
[tex]Z = 3.57[/tex] has a pvalue of 0.9998
1 - 0.9998 = 0.0002
0.0002 = 0.02% probability that the maximum safe weight will be exceeded
