Question

Brush your teeth The amount of time Ricardo spends brushing his teeth follows a Normal distribution with unknown mean and standard deviation. Ricardo spends less than $1$ minute brushing his teeth about $40\%$ of the time. He spends more than $2$ minutes brushing his teeth $2\%$ of the time. Use this information to determine the mean and standard deviation of this distribution.

Short answer

Mean,$\mu =1.1087$

Standard deviation,$\sigma =0.4348$

Step-by-step solution

Given information

Ricardo brushes his teeth for less than a minute roughly $40\%$ of the time.

Ricardo brushes his teeth for more than $2$ minutes $2$ percent of the time.

Calculation

Ricardo spends less than $1$ minute about $40\%$ of the time.

Now,

In the normal probability table of the appendix, find the $z-$score that corresponds to a probability of $40\%$ (or $0.40$).

Note that

The probability that comes closest is $0.40129$which is found in row $-0.2$ and column$.05$ of the normal probability table.

Then

The corresponding $z-$ score,

$z=-0.25$

And

Ricardo spends more than $2$ minutes about $2\%$ of the time.

That means

Ricardo spends less than $2$ minutes about $98\%$ of the time.

Calculation

Now,

In the normal probability table of the appendix, find the $z-$score that corresponds to a probability of $98$percent (or $0.98$)

Note that

The probability that comes closest is $0.9798$, which is found in row $2.0$and column$.05$of the normal probability table.

Then

The corresponding $z-$score,

$z=2.05$

Now,

The value will be the mean multiplied by the product of the z − value and the (standard deviation).

$1=x=\mu +z\sigma =\mu -0.25\sigma$

And

$2=x=\mu +z\sigma =\mu +2.05\sigma$

Subtract the above two equations:

$1=2.30\sigma$

Divide both sides by $2.30$:

We have

Standard deviation,

$\sigma =\frac{1}{2.30}=0.4348$

Then

Find the mean:

$$\begin{gathered} \mu =1+0.25\sigma \\ =1+0.25(0.4348) \\ =1.1087 \end{gathered}$$