Researchers are testing a new diagnostic tool designed to identify a certain condition. The null hypothesis of the significance
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Answer:
27.88% probability that a random sample of 5 has a mean between 677 and 693.
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:
The probability that a random sample of 5 has a mean between 677 and 693.
This is the pvalue of Z when X = 693 subtracted by the pvalue of Z when X = 677. So
X = 693
By the Central Limit Theorem
has a pvalue of 0.8186
X = 677
has a pvalue of 0.5398
0.8186 - 0.5398 = 0.2788
27.88% probability that a random sample of 5 has a mean between 677 and 693.