Question

A set of average city temperatures in May are normally distributed with a mean of 20.66 degrees and a standard deviation of 2 degrees. The average temperature of Singapore is 26 degrees

Answer 1

0.9962

Explanation:

A set of average city temperatures in May are normally distributed with a mean of 20.66 degrees and a standard deviation of 2 degrees. The average temperature of Singapore is 26 degrees

What proportion of average city temperatures are lower than that of Singapore?

Solution:

The z score shows the number of standard deviations by which the raw score is above or below the mean. If the raw score is above the mean then the z score is positive but if the raw score is less than the mean then the raw score is negative. The z score is given by:

`z=(x-\mu)/(\sigma)\\\\where\ x=raw\ score, \mu=mean,\sigma=standard\ deviation`

Given that μ = 20.66, σ = $2

For the average temperature of Singapore of 26°, the z score is:

`z=(x-\mu)/(\sigma)=(26-20.66)/(2) =2.67`

From the normal distribution table, P(x < 26) = P(z < 2.67) = 0.9962

Answer 2

0.9962

Explanation:

The objective of this question is to determine the proportion of average city temperature that is lower than that of Singapore.

Suppose X is a random variable that follows a normal distribution:

Then;

`X \sim N ( \mu = 20.66, \sigma = 2)`

`P(X< 26) = P \bigg ( Z < (26 - 20.66)/(2) \bigg)`

`P(X< 26) = P \bigg ( Z < (5.34)/(2) \bigg)`

`P(X< 26) = P \bigg ( Z < 2.67\bigg)`

From the z tables;

P(X< 26) = 0.9962

Thus, there is 0.9962 proportion of average city temperature are lower than that of Singapore.