a population has mean 15 and standard deviation 5. if 68% of the values in the population are between 10 and 20, the population cannot be normal

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Statement is false, betwen plus/minus 1.96 SD = 68%. Rounding off gives 68% between 10-20.

To verify if the population is normal, we can use a rule called the 68-95-99.7 rule (also known as the empirical rule or the three-sigma rule) which applies to normally distributed data. According to this rule:

- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.

Given that 68% of the values in the population are between 10 and 20, we can conclude that these values fall within one standard deviation of the mean. This means that the mean ± one standard deviation covers 68% of the data.

Now, let's calculate the range of values that lies within one standard deviation from the mean:

Lower Bound = Mean - Standard Deviation = 15 - 5 = 10
Upper Bound = Mean + Standard Deviation = 15 + 5 = 20

We see that the range from 10 to 20 includes 68% of the data, which aligns with the given information. Therefore, we can conclude that the population in question follows a normal distribution.

If the population did not satisfy the criteria of having 68% of the data falling within one standard deviation of the mean, it would suggest a non-normal distribution.