texelaare

2021-08-21

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

smallq9

"Table A contains the probabilities to the left of a z-score or for values smaller than z-score
The time is less than one minute about 40% of the time.
The time is more than two minutes about 2% of the time and thus less than two minutes about 98% of the time.
The z-value is then the value in the table A corresponding to a probability of $\frac{0.40}{0.98}$ :
z=-0.25
z= 2.05
The value is then the mean increased by the product of the z-value and the standard deviation.
$1=x=\mu +z\sigma =\mu -0.25\sigma$
$2=x=\mu +z\sigma =\mu +2.05\sigma$
Subtract the equations:
$1=2.30\sigma$
Divide each side by 2.30:
$0.4348=\frac{1}{2.30}=\sigma$
Determine the mean:
$\mu =1+0.25\sigma =1+0.25\left(0.4348\right)=1.1087$
Result:$\mu =1.1087.\sigma =0.4348$

Do you have a similar question?