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.
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To determine the mean and standard deviation of the distribution, we need to use the properties of the normal distribution and the given information.
Let's denote the mean as μ and the standard deviation as σ.
According to the given information, Ricardo spends less than one minute brushing his teeth 40% of the time. In terms of the standard normal distribution, this can be represented as P(X < 1) = 0.40, where X is a random variable following the standard normal distribution.
Using a standard normal table or a statistical software, we can find the z-score corresponding to a cumulative probability of 0.40, which is approximately -0.25.
We know that the formula for converting a value from a standard normal distribution to a value from an arbitrary normal distribution is: Z = (X - μ) / σ, where Z is the z-score, X is the observed value, μ is the mean, and σ is the standard deviation.
In this case, since we are interested in the value when X = 1, we have:
-0.25 = (1 - μ) / σ
Similarly, using the other given information, we can find the z-score corresponding to a cumulative probability of 0.98, which is approximately 2.05. This represents spending more than two minutes brushing his teeth, which we can denote as P(X > 2) = 0.02.
Using the same formula, we can write:
2.05 = (2 - μ) / σ
Now we have two equations with two unknowns (μ and σ). We can solve this system of equations to find the mean and standard deviation.
Simplifying the first equation:
-0.25σ = 1 - μ
Simplifying the second equation:
2.05σ = 2 - μ
Now we can solve this system of equations. Subtracting the first equation from the second equation:
2.05σ - (-0.25σ) = 2 - 1 - μ + μ
2.30σ = 1
Dividing both sides by 2.30:
σ ≈ 1 / 2.30
σ ≈ 0.4348
We can now substitute this value of σ into either of the original equations to solve for μ. Let's use the first equation:
-0.25(0.4348) = 1 - μ
-0.1087 = 1 - μ
μ ≈ 1 - (-0.1087)
μ ≈ 1.1087
Therefore, the mean (μ) of the distribution is approximately 1.1087 minutes, and the standard deviation (σ) is approximately 0.4348 minutes.