A set of average city temperatures in July are normally distributed with a mean of $23.5^{\circ} \mathrm{C}$ and a standard deviation of $2^{\circ} \mathrm{C}$. The average temperature of Rabat is $23^{\circ} \mathrm{C}$.

What proportion of average city temperatures are higher than that of Rabat?
You may round your answer to four decimal places.

Question

A set of average city temperatures in July are normally distributed with a mean of $23.5^{\circ} \mathrm{C}$ and a standard deviation of $2^{\circ} \mathrm{C}$. The average temperature of Rabat is $23^{\circ} \mathrm{C}$.

What proportion of average city temperatures are higher than that of Rabat?
You may round your answer to four decimal places.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Calculate the z-score for the given x value. ‖ ‖ step 2 ⋮ The z-score is calculated using the formula $z = \frac{x - \mu}{\sigma}$ where $x$ is the value in question, $\mu$ is the mean, and $\sigma$ is the standard deviation. ‖ ‖ step 3 ⋮ For $x = 23$, $\mu = 23.5$, and $\sigma = 2$, the z-score is $z = \frac{23 - 23.5}{2} = -0.25$. ‖ ‖ step 4 ⋮ Use a standard normal distribution table or calculator to find the probability that $Z > -0.25$. ‖ ‖ step 5 ⋮ The probability that $Z > -0.25$ corresponds to the proportion of city temperatures higher than that of Rabat. ‖ ‖ step 6 ⋮ This probability is $0.5 + P(0 < Z < 0.25)$, since the total area under the normal curve is 1 and the curve is symmetric about the mean. ‖ ‖ step 7 ⋮ Using standard normal distribution tables or a calculator, $P(0 < Z < 0.25)$ is approximately 0.0987. ‖ ‖ step 8 ⋮ Therefore, the proportion of city temperatures higher than that of Rabat is $0.5 + 0.0987 = 0.5987$. ‖ ‖ step 9 ⋮ Round the answer to four decimal places as requested. ‖ ∻Answer∻ ⚹ 0.5987 ⚹ ∻Key Concept∻ ⚹Calculating the proportion of values above a given point in a normal distribution using the z-score⚹ ∻Explanation∻ ⚹The z-score represents the number of standard deviations a value is from the mean. The area to the right of the z-score in a standard normal distribution gives the proportion of values above that value.⚹◊What is the formula for converting a z-score into a percentile in a normal distribution?⍭ Generate me a similar question◊

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