A botanist was measuring how tall her plant grew. After two weeks it had grown 12.93 inches.The second week alone it had grown 3
1 answer:
You might be interested in
Answer:
a. 48.80%
b. 46.02%
c. 57.93% probability of the sample mean weight being above the weight limit, which is a high probability, meaning that the elevator does not appear to have the correct weight limit
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 normally distributed samples are 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 normally distributed random variable X, with mean and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this problem, we have that:
a. find the probability that if a person is randomly selected, his weight will be greater than 176 pounds.
This is 1 subtracted by the pvalue of Z when X = 176. So
has a pvalue of 0.5120
1 - 0.5120 = 0.4880
48.80% probability that if a person is randomly selected, his weight will be greater than 176 pounds.
b. Find the probability that 10 randomly selected people will have a neam that is greater than 176 pounds.
Now
This is 1 subtracted by the pvalue of Z when X = 176. So
By the Central Limit Thorem
has a pvalue of 0.5398
1 - 0.5398 = 0.4602
46.02% probability that 10 randomly selected people will have a neam that is greater than 176 pounds.
c. Does the elevator appear to have the correct weight limit?
The weight limit is 173 pounds in a sample of 10.
The probability that the mean weight is larger than this is 1 subtracted by the pvalue of Z when X = 173. So
has a pvalue of 0.4207
1 - 0.4207 = 0.5793
57.93% probability of the sample mean weight being above the weight limit, which is a high probability, meaning that the elevator does not appear to have the correct weight limit