In a set of 400 values, the mean is 121 and the standard deviation is 14. Estimate how many values fall in between each interval?

a) 107 - 135
b)93 - 149
c) 79 - 163

107 = µ-1σ

135 = µ+1σ
So, what % is within 1σ of µ? Your Z table says 68.27% of 400 or 273

Treat the other numbers similarly

To estimate how many values fall in between each interval, we can use the empirical rule or the 68-95-99.7 rule.

The empirical rule states that for a normal distribution:
- 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

Using this rule, we can estimate the number of values falling within each interval.

a) For the interval 107-135: This range is within one standard deviation of the mean (121 ± 14). According to the empirical rule, approximately 68% of the data falls within this range. Therefore, we can estimate that about 68% of the 400 values, which is equal to 0.68 * 400 = 272 values, fall within the interval of 107-135.

b) For the interval 93-149: This range is within two standard deviations of the mean (121 ± 2*14). According to the empirical rule, approximately 95% of the data falls within this range. Therefore, we can estimate that about 95% of the 400 values, which is equal to 0.95 * 400 = 380 values, fall within the interval of 93-149.

c) For the interval 79-163: This range is within three standard deviations of the mean (121 ± 3*14). According to the empirical rule, approximately 99.7% of the data falls within this range. Therefore, we can estimate that about 99.7% of the 400 values, which is equal to 0.997 * 400 = 398.8 (approximately 399) values, fall within the interval of 79-163.

Please note that these are estimates based on the empirical rule, assuming that the data follows a normal distribution. The actual number of values falling within each interval may vary.