Share
Suppose that the walking step lengths of adult males are normally distributed with a mean of 2.5 feet and a standard deviation of 0.2 feet.
Question
Suppose that the walking step lengths of adult males are normally distributed with a mean of 2.5 feet and a standard deviation of 0.2 feet. A sample of 41 men’s step
lengths is taken
Step 2 of 2: Find the probability that the mean of the sample taken is less than 2.2 feet. Round your answer to 4 decimal places, if necessary,
in progress
0
Mathematics
4 years
2021-08-17T07:47:56+00:00
2021-08-17T07:47:56+00:00 1 Answers
14 views
0
Answers ( )
Answer:
0% probability that the mean of the sample taken is less than 2.2 feet.
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
and standard deviation
, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem establishes 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.
Mean of 2.5 feet and a standard deviation of 0.2 feet.
This means that
Sample of 41
This means that
Find the probability that the mean of the sample taken is less than 2.2 feet.
This is the p-value of Z when X = 2.2 So
By the Central Limit Theorem
0% probability that the mean of the sample taken is less than 2.2 feet.