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The mean high temperature on a particular day in January is 31 degrees F and the standard deviation is 8.7 degrees. One year, the temperature was 52 degrees F on that day. What is the Z-score for that day's temperature? Round to the appropriate number of decimal places for Z-scores. Is this temperature significantly high? (greater than 2)​

User PeteShack
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2 votes

Answer:

The Z-score for that day's temperature is 2.41, and since Z > 2, this temperature is significantly high.

Explanation:

Z-score:

In a set with mean
\mu and standard deviation
\sigma, the z-score of a measure X is given by:


Z = (X - \mu)/(\sigma)

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.

If
|Z| > 2, the measure X is significantly high(Z > 2) or significantly low(Z < -2).

The mean high temperature on a particular day in January is 31 degrees F and the standard deviation is 8.7 degrees.

This means that
\mu = 31, \sigma = 8.7

One year, the temperature was 52 degrees F on that day.

This means that
X = 52.

What is the Z-score for that day's temperature?


Z = (X - \mu)/(\sigma)


Z = (52 - 31)/(8.7)


Z = 2.41

The Z-score for that day's temperature is 2.41, and since Z > 2, this temperature is significantly high.

User Yong Wang
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