Question

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the mean midday temperature recorded in june in a city in south california is 36°c, and the standard deviation is 3°c. the number of days in the month of june is 30.
assuming the data is normally distributed, the number of days in june when the midday temperature was between 39°c and 42°c is .

Explanation:

Step1: Identify z - scores

The formula for the z - score is $z=\frac{x - \mu}{\sigma}$, where $\mu$ is the mean, $\sigma$ is the standard deviation, and $x$ is the value from the dataset.

Given $\mu = 36^{\circ}C$, $\sigma=3^{\circ}C$.

For $x = 39^{\circ}C$:
$z_1=\frac{39 - 36}{3}=\frac{3}{3}=1$

For $x = 42^{\circ}C$:
$z_2=\frac{42 - 36}{3}=\frac{6}{3}=2$

Step2: Find the area between z - scores

We know that for a normal distribution:

• The area to the left of $z = 1$ is $P(Z\lt1)=0.8413$ (from standard normal distribution table)
• The area to the left of $z = 2$ is $P(Z\lt2)=0.9772$ (from standard normal distribution table)

The area between $z = 1$ and $z = 2$ is $P(1\lt Z\lt2)=P(Z\lt2)-P(Z\lt1)=0.9772 - 0.8413 = 0.1359$

Step3: Calculate the number of days

The total number of days in June is $n = 30$.

The number of days with temperature between $39^{\circ}C$ and $42^{\circ}C$ is $N=n\times P(1\lt Z\lt2)=30\times0.1359\approx4.077\approx4$ (we round to the nearest whole number since we are counting days)

Answer:

4