Questions: The standard normal probability distribution has a mean of 0 and a standard deviation of 1

Transcript text: The standard normal probability distribution has a mean of $\square$ and a standard deviation of $\square$

Solution

Solution Steps

Step 1: Understanding the Standard Normal Distribution

The standard normal probability distribution is a specific case of the normal distribution characterized by a mean ($ \mu $) of $ 0 $ and a standard deviation ($ \sigma $) of $ 1 $. This means that the distribution is centered at $ 0 $ and the spread of the data is determined by the standard deviation.

Step 2: Calculating the Probability

To find the probability that a sample mean falls within one standard deviation of the mean, we calculate the Z-scores for the range from $ -1 $ to $ 1 $:

$ Z_{start} = -1.0 $
$ Z_{end} = 1.0 $

Using the cumulative distribution function $ \Phi $, the probability $ P $ that the sample mean falls within this range is given by: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(1.0) - \Phi(-1.0) = 0.6827 \]

Step 3: Summary of Results

The results indicate that approximately $ 68.27\% $ of the data in a standard normal distribution lies within one standard deviation of the mean.

Final Answer

The standard normal probability distribution has a mean of $ \boxed{0} $ and a standard deviation of $ \boxed{1} $. The probability that the sample mean falls within one standard deviation of the mean is $ \boxed{0.6827} $.

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