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Scorpion4ik [409]
3 years ago
6

The amount of time Ricardo spends brushing his teeth follows a Normal distribution with unknown mean and standard deviation. Ric

ardo 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
Mathematics
1 answer:
Papessa [141]3 years ago
7 0
Let X denote the number of minutes spent brushing his teeth, and let \mu be the mean and \sigma the standard deviation for this distribution.

\mathbb P(X

The z-score corresponding to this probability is approximately z=-0.2533, which means

\dfrac{1-\mu}\sigma=-0.2533\iff\mu-0.2533\sigma=1

Next, (note the sign change)

\mathbb P(X>2)=0.02\implies\mathbb P(X\le2)=\mathbb P\left(\dfrac{X-\mu}\sigma\le\dfrac{2-\mu}\sigma\right)=0.98

The corresponding z-score is approximately z=2.0538, so you have

\dfrac{2-\mu}\sigma=2.0538\iff\mu+2.0538\sigma=2

Solving the two equations for \mu and \sigma, you'll find that the mean is approximately \mu=1.1098 and the standard deviation is approximately \sigma=0.4334.
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