On Tuesday September 27, 2016 the high temperature increased to 36.7 degrees Celsius (98 degrees Fahrenheit). What is the probability that we observe a temperature of 36.7 degrees Celsius or higher given that the average temperature is 27 degrees Celsius with a standard deviation of 2.78 degrees Celsius? Provide your answer to four decimal places.
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To find the probability that we observe a temperature of 36.7 degrees Celsius or higher, given the average temperature and standard deviation, we need to use the concept of standard deviation and the normal distribution.
First, we need to standardize the temperature of 36.7 degrees Celsius using the formula for z-score:
z = (x - μ) / σ
where z is the z-score, x is the temperature, μ is the mean, and σ is the standard deviation.
Using the given information:
x = 36.7 degrees Celsius
μ = 27 degrees Celsius
σ = 2.78 degrees Celsius
Substituting these values into the formula:
z = (36.7 - 27) / 2.78
Calculating:
z ≈ 3.48
Now, we need to find the probability associated with this z-score using the standard normal distribution table or a statistical calculator. The cumulative probability (often denoted as P(Z > z)) is the probability that a random variable from the standard normal distribution is greater than z.
Looking up the z-score of 3.48 in a standard normal distribution table or using a statistical calculator, we find that the cumulative probability is approximately 0.9997.
Therefore, the probability of observing a temperature of 36.7 degrees Celsius or higher, given the average temperature and standard deviation, is approximately 0.9997.