For the normal distribution, the mean plus and minus 1.96 standard deviations will include about what percent of the observations?
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A. B. C. D. E.D
95% of the are under the curve will lie within plus and minus 2 standard deviations of the mean.
The correct answer is D. 95%.
In a normal distribution, which is also known as a bell curve, the mean (average) and standard deviation play important roles in describing the distribution of the data. The mean represents the central tendency, while the standard deviation measures the dispersion or spread of the data points.
In a standard normal distribution, which has a mean of 0 and a standard deviation of 1, the area under the curve between two specific z-scores represents the proportion or percentage of observations falling within that range. The z-score is a measure of how many standard deviations a particular observation is from the mean.
In the case of the question, we are asked about the range that includes the mean plus and minus 1.96 standard deviations. The value of 1.96 is the critical value that corresponds to a 95% confidence level. This means that approximately 95% of the observations in a normal distribution will fall within this range.
To see why this is the case, we can refer to the empirical rule, also known as the 68-95-99.7 rule, which describes the percentage of observations falling within certain standard deviation ranges in a normal distribution:
Since we are considering the range of plus and minus 1.96 standard deviations, which is within two standard deviations, we can conclude that approximately 95% of the observations will be included in this range.
Therefore, the correct answer is D. 95%.