Final answer:
At least 75% of the observations lie between 21 and 29. While at least 95% of the observations lie within 4.5 standard deviations, we can only ascertain that the percentage of observations that lie between 17 and 33 is at least 95% using Chebyshev's rule.
Step-by-step explanation:
The question requires the application of Chebyshev's rule to a quantitative data set with a mean of 25 and a standard deviation of 2. According to Chebyshev's rule, at least 75% of the observations lie within two standard deviations of the mean.
Part a
To find the range of values where at least 75% of the observations lie, we calculate:
- Lower bound: 25 - (2 x 2) = 21
- Upper bound: 25 + (2 x 2) = 29
Therefore, at least 75% of the observations lie between 21 and 29.
Part b
For the second part, we look at the range between 17 and 33, which is 4 standard deviations from the mean (since 33 - 25 = 8 and 25 - 17 = 8, and 8 is 4 times the standard deviation of 2).
According to Chebyshev's rule, at least 95% of the observations will lie within 4.5 standard deviations of the mean. Since 4 is less than 4.5, we can be confident that at least 95% of the observations fall within this range. However, without assuming the distribution is normal (bell-shaped), we cannot specify the exact percentage beyond saying it is at least 95%. If the distribution was normal, we could use the Empirical rule to say approximately 95%.