Question

Suppose that IQ scores have a bell-shaped distribution with a mean of 103 and a standard deviation of 13. Describe where the highest and lowest 5 % of IQ scores lie. Answer Keypad Keyboard Shortcut O between 77 and 129 O below 64 and above 142 below 77 and above 129 O below 90 and above 116

Answer 1

Between 77 and 129

Here, we want to describe where the lowest 5% and the highest 5% of IQ scores lie

What we need here is the appropriate z-score that corresponds to both the lowest 5% and the highest 5%

For the lowest 5%, we have a z-score of -1.645 while for the highest 5% (95% - 100%), we have a z-score of 1.645

Mathematically, the formula for calculating the z-score for a normal distribution is as follows;

`\text{Z}_{score\text{ }}=\text{ }\frac{(x\text{ - }\mu)}{\sigma}`

Where σ is the standard deviation = 13 and μ = 103 which is the mean. X refer to the raw IQ scores which we are trying to calculate

Thus;

for z = -1.645

`\begin{gathered} -1.645\text{ = }((x-103))/(13) \\ \\ 13(-1.645)\text{ = x-103} \\ \\ -21.385\text{ = x -103} \\ x\text{ = 103 - 21.385} \\ \\ x\text{ = 81.615 } \\ \\ \text{approx. 82} \end{gathered}`

For z = 1.645

`\begin{gathered} 1.645\text{ = }((x-103))/(13) \\ \\ x-103\text{ = 13(1.645)} \\ \\ x-103\text{ = 21.385} \\ \\ x\text{ = 21.385 + 103} \\ x\text{ = 124.385} \\ \\ \text{approx 124} \end{gathered}`

So we have the range of between 82 and 124

We now proceed to the options to check the best fit

The best fit here is between 77 and 129