McClass

Question

What is the probability of having realizations below $129 ?$

Calculate the $z$ -value and look up the corresponding $φ$ -value.
To calculate the $z$ -value, use $\frac{X - μ}{σ} .$
Use the lower tail of the normal distribution and select the value corresponding to the $z$ -value.

Feedback

Waiting for response...

Solution(s)

The system accepts the following correct response(s)

$z = \frac{X - μ}{σ}$
$p = Φ (z)$

You correctly identified the main steps of the calculation.
You correctly calculated the $z$ -value.
You calculated the correct tail of the distribution.

Incorrect responses

The system detects the following incorrect responses

$z = \frac{\underset{instead of X - μ}{\underset{⏟}{X + μ}}}{σ}$
$p = \underset{instead of Φ (z)}{\underset{⏟}{1 - Φ (z)}}$

You correctly identified the main steps of the calculation.
You correctly calculated the $z$ -value.
Subtract $μ$ from $X$ instead of adding them up.
Your answer matches the $wrong tail$ of the normal distribution.

$z = \frac{X - μ}{σ}$
$p = \underset{instead of Φ (z)}{\underset{⏟}{1 - Φ (z)}}$

You correctly identified the main steps of the calculation.
You correctly calculated the $z$ -value.
Your answer matches the $wrong tail$ of the normal distribution.

$z = \frac{X - \underset{instead of μ}{\underset{⏟}{σ}}}{\underset{instead of σ}{\underset{⏟}{μ}}}$
$p = \underset{instead of Φ (z)}{\underset{⏟}{1 - Φ (z)}}$

You swapped $μ$ and $σ .$
You correctly identified the main steps of the calculation.
You correctly calculated the $z$ -value.
Your answer matches the $wrong tail$ of the normal distribution.

$z = \frac{\underset{instead of X - μ}{\underset{⏟}{X + μ}}}{\underset{instead of σ}{\underset{⏟}{σ^{2}}}}$
$p = \underset{instead of Φ (z)}{\underset{⏟}{1 - Φ (z)}}$

You squared $σ$ by mistake.
You correctly identified the main steps of the calculation.
You correctly calculated the $z$ -value.
Subtract $μ$ from $X$ instead of adding them up.
Your answer matches the $wrong tail$ of the normal distribution.

$z = \frac{X - μ}{\underset{instead of σ}{\underset{⏟}{σ^{2}}}}$
$p = \underset{instead of Φ (z)}{\underset{⏟}{1 - Φ (z)}}$

You squared $σ$ by mistake.
You correctly identified the main steps of the calculation.
You correctly calculated the $z$ -value.
Your answer matches the $wrong tail$ of the normal distribution.

$z = \frac{X - μ}{\underset{instead of σ}{\underset{⏟}{σ^{2}}}}$
$p = Φ (z)$

You squared $σ$ by mistake.
You correctly identified the main steps of the calculation.
You correctly calculated the $z$ -value.
You calculated the correct tail of the distribution.

$z = \frac{\underset{instead of X - μ}{\underset{⏟}{X + μ}}}{\underset{instead of σ}{\underset{⏟}{σ^{2}}}}$
$p = Φ (z)$

You squared $σ$ by mistake.
You correctly identified the main steps of the calculation.
You correctly calculated the $z$ -value.
Subtract $μ$ from $X$ instead of adding them up.
You calculated the correct tail of the distribution.

$z = \frac{X - \underset{instead of μ}{\underset{⏟}{σ}}}{\underset{instead of σ}{\underset{⏟}{μ}}}$
$p = Φ (z)$

You swapped $μ$ and $σ .$
You correctly identified the main steps of the calculation.
You correctly calculated the $z$ -value.
You calculated the correct tail of the distribution.

$z = \frac{\underset{instead of X - μ}{\underset{⏟}{X + μ}}}{σ}$
$p = Φ (z)$

You correctly identified the main steps of the calculation.
You correctly calculated the $z$ -value.
Subtract $μ$ from $X$ instead of adding them up.
You calculated the correct tail of the distribution.

$z = \frac{X - μ}{σ}$
$p = \underset{instead of Φ (z)}{\underset{⏟}{z / 100}}$

You correctly identified the main steps of the calculation.
You correctly calculated the $z$ -value.
The $z$ -value is calculated by using the formula $\frac{X - μ}{σ} .$
You mistakenly divided the $z$ -value by 100 instead of retrieving the value from the normal distribution.

Traps

Avoid these traps

Do not use the upper tail of the normal distribution.
Do not divide the $z$ -value by 100, use the normal distribution instead.
Remember to calculate the $z$ -value.
Try using $μ$ and $σ$ in a different configuration.
Try using subtraction rather than addition in $X + μ .$
Use $σ$ instead of $σ^{2} .$

Question

Which quantile marks the upper $43 %?$

First determine the $z$ -statistic, then translate it to the original scale.
Use the lower tail of the normal distribution and select the value corresponding to the $z$ -value.

Feedback

Waiting for response...

Solution(s)

The system accepts the following correct response(s)

$z = Φ^{- 1} (1 - \frac{p}{100})$
$X = z σ + μ$

Correctly determined the $z$ -statistic and translated it to the original scale.
Correctly used the lower tail of the normal distribution.

Incorrect responses

The system detects the following incorrect responses

$z = Φ^{- 1} (\underset{instead of 1 - \frac{p}{100}}{\underset{⏟}{\frac{p}{10}}})$
$X = \underset{instead of z σ + μ}{\underset{⏟}{z}}$

Correctly determined the $z$ -statistic and translated it to the original scale.
Your response matches the lower instead of the upper tail of the normal distribution.
You divided the percentage by $10$ instead of 100 to obtain a proportion.
You only calculated the $z$ -value.

$z = Φ^{- 1} (\underset{instead of 1 - \frac{p}{100}}{\underset{⏟}{\frac{p}{10}}})$
$X = z σ + μ$

Correctly determined the $z$ -statistic and translated it to the original scale.
Your response matches the lower instead of the upper tail of the normal distribution.
You divided the percentage by $10$ instead of 100 to obtain a proportion.

$z = Φ^{- 1} (\underset{instead of 1 - \frac{p}{100}}{\underset{⏟}{\frac{p}{10}}})$
$X = \underset{instead of z σ + μ}{\underset{⏟}{z μ + σ}}$

Correctly determined the $z$ -statistic and translated it to the original scale.
Your response matches the lower instead of the upper tail of the normal distribution.
You divided the percentage by $10$ instead of 100 to obtain a proportion.
You swapped $μ$ and $σ .$

$z = Φ^{- 1} (\underset{instead of 1 - \frac{p}{100}}{\underset{⏟}{\frac{p}{10}}})$
$X = z σ^{2} + μ$

Correctly determined the $z$ -statistic and translated it to the original scale.
Your response matches the lower instead of the upper tail of the normal distribution.
You divided the percentage by $10$ instead of 100 to obtain a proportion.
You squared $σ$ by mistake.

$z = Φ^{- 1} (1 - \underset{instead of \frac{p}{100}}{\underset{⏟}{\frac{p}{10}}})$
$X = \underset{instead of z σ + μ}{\underset{⏟}{z}}$

Correctly determined the $z$ -statistic and translated it to the original scale.
Correctly used the lower tail of the normal distribution.
You divided the percentage by $10$ instead of 100 to obtain a proportion.
You only calculated the $z$ -value.

$z = Φ^{- 1} (1 - \underset{instead of \frac{p}{100}}{\underset{⏟}{\frac{p}{10}}})$
$X = z σ + μ$

Correctly determined the $z$ -statistic and translated it to the original scale.
Correctly used the lower tail of the normal distribution.
You divided the percentage by $10$ instead of 100 to obtain a proportion.

$z = Φ^{- 1} (1 - \underset{instead of \frac{p}{100}}{\underset{⏟}{\frac{p}{10}}})$
$X = \underset{instead of z σ + μ}{\underset{⏟}{z μ + σ}}$

Correctly determined the $z$ -statistic and translated it to the original scale.
Correctly used the lower tail of the normal distribution.
You divided the percentage by $10$ instead of 100 to obtain a proportion.
You swapped $μ$ and $σ .$

$z = Φ^{- 1} (1 - \underset{instead of \frac{p}{100}}{\underset{⏟}{\frac{p}{10}}})$
$X = z σ^{2} + μ$

Correctly determined the $z$ -statistic and translated it to the original scale.
Correctly used the lower tail of the normal distribution.
You divided the percentage by $10$ instead of 100 to obtain a proportion.
You squared $σ$ by mistake.

$z = Φ^{- 1} (\underset{instead of 1 - \frac{p}{100}}{\underset{⏟}{\frac{p}{\underset{instead of 100}{\underset{⏟}{1000}}}}})$
$X = z σ^{2} + μ$

Correctly determined the $z$ -statistic and translated it to the original scale.
Your response matches the lower instead of the upper tail of the normal distribution.
You divided the percentage by $1000$ instead of 100 to obtain a proportion.
You squared $σ$ by mistake.

$z = Φ^{- 1} (\underset{instead of 1 - \frac{p}{100}}{\underset{⏟}{\frac{p}{\underset{instead of 100}{\underset{⏟}{1000}}}}})$
$X = \underset{instead of z σ + μ}{\underset{⏟}{z μ + σ}}$

Correctly determined the $z$ -statistic and translated it to the original scale.
Your response matches the lower instead of the upper tail of the normal distribution.
You divided the percentage by $1000$ instead of 100 to obtain a proportion.
You swapped $μ$ and $σ .$

$z = Φ^{- 1} (\underset{instead of 1 - \frac{p}{100}}{\underset{⏟}{\frac{p}{100}}})$
$X = \underset{instead of z σ + μ}{\underset{⏟}{z μ + σ}}$

Correctly determined the $z$ -statistic and translated it to the original scale.
Your response matches the lower instead of the upper tail of the normal distribution.
You swapped $μ$ and $σ .$

$z = Φ^{- 1} (\underset{instead of 1 - \frac{p}{100}}{\underset{⏟}{\frac{p}{\underset{instead of 100}{\underset{⏟}{1000}}}}})$
$X = \underset{instead of z σ + μ}{\underset{⏟}{z}}$

Correctly determined the $z$ -statistic and translated it to the original scale.
Your response matches the lower instead of the upper tail of the normal distribution.
You divided the percentage by $1000$ instead of 100 to obtain a proportion.
You only calculated the $z$ -value.

$z = Φ^{- 1} (\underset{instead of 1 - \frac{p}{100}}{\underset{⏟}{\frac{p}{100}}})$
$X = \underset{instead of z σ + μ}{\underset{⏟}{z}}$

Correctly determined the $z$ -statistic and translated it to the original scale.
Your response matches the lower instead of the upper tail of the normal distribution.
You only calculated the $z$ -value.

$z = Φ^{- 1} (1 - \frac{p}{100})$
$X = \underset{instead of z σ + μ}{\underset{⏟}{z}}$

Correctly determined the $z$ -statistic and translated it to the original scale.
Correctly used the lower tail of the normal distribution.
You only calculated the $z$ -value.

$z = Φ^{- 1} (1 - \frac{p}{\underset{instead of 100}{\underset{⏟}{1000}}})$
$X = \underset{instead of z σ + μ}{\underset{⏟}{z}}$

Correctly determined the $z$ -statistic and translated it to the original scale.
Correctly used the lower tail of the normal distribution.
You divided the percentage by $1000$ instead of 100 to obtain a proportion.
You only calculated the $z$ -value.

$z = Φ^{- 1} (1 - \frac{p}{100})$
$X = \underset{instead of z σ + μ}{\underset{⏟}{z μ + σ}}$

Correctly determined the $z$ -statistic and translated it to the original scale.
Correctly used the lower tail of the normal distribution.
You swapped $μ$ and $σ .$

$z = Φ^{- 1} (\underset{instead of 1 - \frac{p}{100}}{\underset{⏟}{\frac{p}{100}}})$
$X = z σ^{2} + μ$

Correctly determined the $z$ -statistic and translated it to the original scale.
Your response matches the lower instead of the upper tail of the normal distribution.
You squared $σ$ by mistake.

$z = Φ^{- 1} (\underset{instead of 1 - \frac{p}{100}}{\underset{⏟}{\frac{p}{\underset{instead of 100}{\underset{⏟}{1000}}}}})$
$X = z σ + μ$

Correctly determined the $z$ -statistic and translated it to the original scale.
Your response matches the lower instead of the upper tail of the normal distribution.
You divided the percentage by $1000$ instead of 100 to obtain a proportion.

$z = Φ^{- 1} (\underset{instead of 1 - \frac{p}{100}}{\underset{⏟}{\frac{p}{100}}})$
$X = z σ + μ$

Correctly determined the $z$ -statistic and translated it to the original scale.
Your response matches the lower instead of the upper tail of the normal distribution.

$z = Φ^{- 1} (1 - \frac{p}{\underset{instead of 100}{\underset{⏟}{1000}}})$
$X = z σ + μ$

Correctly determined the $z$ -statistic and translated it to the original scale.
Correctly used the lower tail of the normal distribution.
You divided the percentage by $1000$ instead of 100 to obtain a proportion.

$z = Φ^{- 1} (1 - \frac{p}{100})$
$X = z σ^{2} + μ$

Correctly determined the $z$ -statistic and translated it to the original scale.
Correctly used the lower tail of the normal distribution.
You squared $σ$ by mistake.

$z = Φ^{- 1} (1 - \frac{p}{\underset{instead of 100}{\underset{⏟}{1000}}})$
$X = \underset{instead of z σ + μ}{\underset{⏟}{z μ + σ}}$

Correctly determined the $z$ -statistic and translated it to the original scale.
Correctly used the lower tail of the normal distribution.
You divided the percentage by $1000$ instead of 100 to obtain a proportion.
You swapped $μ$ and $σ .$

$z = Φ^{- 1} (1 - \frac{p}{\underset{instead of 100}{\underset{⏟}{1000}}})$
$X = z σ^{2} + μ$

Correctly determined the $z$ -statistic and translated it to the original scale.
Correctly used the lower tail of the normal distribution.
You divided the percentage by $1000$ instead of 100 to obtain a proportion.
You squared $σ$ by mistake.

Traps

Avoid these traps

Do not select the upper tail of the normal distribution.
Make sure to divide by 100 when translating a percentage to a proportion.
Translate the $z$ -value back to the original scale.
Try using $μ$ and $σ$ in a different configuration.
Use $σ$ instead of $σ^{2} .$

Normal distribution

Question

0.97

0.00

0.03

0.13

0.15

0.45

0.55

0.85

0.87

1.00

1.29

Question

102.65

nan

nan

nan

nan

nan

nan

nan

nan

-286.30

-156.69

-2.64

-1.72

-0.18

0.18

1.72

32.64

60.32

74.25

97.35

125.75

139.68

186.69

486.30