Let's break down and analyze the problems you have shared:

Part (f): Range of Variation

The range of variation is given by the interval $[ \bar{X} - S, \bar{X} + S ]$, where:

• $\bar{X}$ is the mean
• $S$ is the standard deviation

The correct results are:

• Lower bound: $\bar{X} - S = 0.98$
• Upper bound: $\bar{X} + S = 5.62$

These values were correctly entered in the system, so no issues there.

Part (g): Percent of Data Values within One Standard Deviation

The question asks to calculate the percentage of data values that fall within one standard deviation from the mean, i.e., within the interval $[ \bar{X} - S, \bar{X} + S ]$.

The total area of the shaded region corresponds to the probability mass (or percentage) of the data values that lie within this range for a normally distributed dataset. Based on the empirical rule for a normal distribution:

• Approximately 68.27% of the data falls within one standard deviation from the mean.

You entered "25" as the total area, which seems incorrect. The correct approach would involve multiplying the total shaded area by 100 to express it as a percentage of the entire dataset.

If this is a normal distribution, the correct value is approximately 68.27%, not 25%.

Would you like further details or help on this?

Related Questions:

1. How can we interpret the empirical rule (68-95-99.7) in a real dataset?
2. What steps are involved in calculating the percentage area under the normal curve?
3. How is the range of variation applied in statistical data analysis?
4. Why does the normal distribution help with calculating probabilities?
5. What happens when the data is not normally distributed?

Tip:

When dealing with a normal distribution, remember the empirical rule: 68.27% of data falls within one standard deviation, 95.45% within two, and 99.73% within three.