Answer:
There is a 49.20% probability that a single randomly selected value is less than 148.3.
There is a 38.21% probability that a sample of size n = 186 n=186 is randomly selected with a mean less than 148.3.
Step-by-step explanation:
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.
In this problem, we have that:
A population of values has a normal distribution with and
.
Find the probability that a single randomly selected value is less than 148.3
This is the pvalue of Z when .
has a pvalue of 0.4920
There is a 49.20% probability that a single randomly selected value is less than 148.3.
Find the probability that a sample of size n = 186 n=186 is randomly selected with a mean less than 148.3.
We want to find the mean of the sample, so we have to find the standard deviation of the population. That is
Now, we have to find the pvalue of Z when .
has a pvalue of 0.3821
There is a 38.21% probability that a sample of size n = 186 n=186 is randomly selected with a mean less than 148.3.