Question

1/ Assume that the distribution of residuals is approximately normal with mean 0 cm and standard deviation 5.9cm. What percent of the residuals are greater than 8cm? Justify your answer.
2/ Base on your answer on part 1, would it be surprising to randomly select a high school senior from the high school with a foot length of 20cm and a height greater than 165cm? Justify your answer.

Answer 1

Answer:

1. 8.576%

2. Yes

Step-by-step explanation:

1. Use the calculator

Enter: normalcdf(8, E99, 0, 5.9)

This is equal to .08756.

Convert this number into a percent = 8.756%

2.

First, calculate the residual

Residual = Observed - Predicted = 165 - 2.599(20) + 105.08 = 7.94

Use the calculator

Enter: normalcdf(7.94, E99, 0, 5.9)

This is equal to .0892.

Convert this number into a percent = 8.92%

This would be surprising, because the chance of this happening is very low.

Answer 2

Using the binomial probability concept, the percentage of residuals greater than 8cm is 8.75% and thus randomly selecting a senior with a residual below 8 cm won't be surprising.

Recall :

• Zscore = (X - mean) ÷ standard deviation

Percentage of residual greater than 8cm:

• Zscore = (8 - 0) / 5.9 = 1.356

Using a normal distribution table :

P(Z > 1.356) = 1 - P(Z < 1.356)

P(Z > 1.356) = 1 - 0.91245 = 0.0875 = 8.75%

The prediction equation :

y = 2.599(x) + 105.08

y = 2.599(20) + 105.08

y = 157.06

Residual = Actual - Predicted

Residual = 165 - 157.06 = 7.94

Since the residual is less than 8cm and the probability of selecting a high school senior with a residual below 8cm is about 91.25% ; Hence, it would not be surprising.

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