Solution - Cumulative probability in the standard normal distribution
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Cumulative probability in the standard normal distributionStep-by-step explanation
1. Find the cumulative probability of the z-scores values up to
More than of the time, data with a standard normal distribution lies within plus or minus standard deviations of the mean.
The cumulative probability of the values up to is .
The cumulative probability that is
2. Find the cumulative probability of the z-scores values up to
More than of the time, data with a standard normal distribution lies within plus or minus standard deviations of the mean.
The cumulative probability of the values up to is .
The cumulative probability that is
3. Calculate the cumulative probability between 293 and 268
To find the cumulative probability of the area between the two z-scores, subtract the smaller cumulative probability (everything to the left of ) from the larger cumulative probability (everything to the left of ):
The cumulative probability that is
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Please leave us feedback.Why learn this
The normal distribution is important because we see it often in nature. Suppose we gather many unrelated measures, like human heights, blood pressure readings, or IQ scores. They will follow the normal distribution.
We see many normally distributed variables in psychology. For example, reading ability, introversion or job satisfaction. In investing, the normal distribution shows asset class returns. Although these distributions are only roughly normal, they are pretty close, and we can treat them as normal.
The normal distribution is easy to work with. Many statistical tests rely on it. Moreover, these tests work well even when the distribution is only approximately normal. For example, if a set's mean and standard deviation are known, and the set follows the normal distribution, we can easily convert between percentiles and raw scores.
Any normal distribution can be standardized to a standard normal distribution. That way, we can compare two or more separate data sets. Using standard normal distribution, we can estimate probabilities of events involving normal distribution. This way, we can estimate how tall a person is likely to grow, for instance.