To determine the probability that a random day in August has a high temperature that exceeds 80°F, we start by considering the necessary statistical data and steps.
1. Data Consideration:
- Mean high temperature for August (μ): 78°F
- Standard deviation of high temperatures (σ): 5°F
- Threshold temperature (T): 80°F
2. Calculate the Z-Score:
The Z-score is a measure of how many standard deviations an element is from the mean. The formula to calculate the Z-score (Z) is:
[tex]\[
Z = \frac{T - μ}{σ}
\][/tex]
Plugging in the values we have:
[tex]\[
Z = \frac{80 - 78}{5} = \frac{2}{5} = 0.4
\][/tex]
3. Interpret the Z-Score:
A Z-score of 0.4 indicates that 80°F is 0.4 standard deviations above the mean temperature.
4. Find the Probability:
To find the probability that the temperature exceeds 80°F, we need to use the cumulative distribution function (CDF) of the standard normal distribution. The CDF tells us the probability that a normally distributed random variable is less than or equal to a certain value, i.e., P(Z ≤ 0.4).
To find P(Z > 0.4), we need to consider that the total area under the normal distribution curve is 1 (representing 100%). Therefore:
[tex]\[
P(Z > 0.4) = 1 - P(Z ≤ 0.4)
\][/tex]
From standard normal distribution tables or using statistical software, we find P(Z ≤ 0.4) ≈ 0.6554.
5. Calculate the Exceedance Probability:
Thus, the probability that the temperature exceeds 80°F is:
[tex]\[
P(Z > 0.4) = 1 - 0.6554 = 0.3446
\][/tex]
6. Round to the Nearest Hundredth:
Finally, rounding 0.3446 to the nearest hundredth gives us 0.34.
Therefore, the probability that a random day in August has a high temperature exceeding 80°F is 0.34 or 34%.
