If a set of observations is normally distributed with mean μ and variance σ², what percent of these differ from the mean by less than 0.52σ?

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Understand the Problem

The question is asking for the percentage of observations in a normally distributed set that differ from the mean by less than 0.52 times the standard deviation. This involves understanding properties of the normal distribution and using the empirical rule or z-scores to determine the answer.

Answer

Approximately $40.04\%$ of observations differ from the mean by less than $0.52\sigma$.
Answer for screen readers

Approximately $40.04%$ of observations differ from the mean by less than $0.52\sigma$.

Steps to Solve

  1. Identify the standard deviation range We need to determine the range of values that differ from the mean by less than $0.52\sigma$. This range can be represented as: $$ \mu - 0.52\sigma < X < \mu + 0.52\sigma $$

  2. Convert to z-scores To find the percentage of observations within this range, we convert the limits to z-scores using the formula: $$ z = \frac{X - \mu}{\sigma} $$ Thus, the z-scores corresponding to $X = \mu - 0.52\sigma$ and $X = \mu + 0.52\sigma$ are: $$ z_1 = \frac{\mu - 0.52\sigma - \mu}{\sigma} = -0.52 $$ $$ z_2 = \frac{\mu + 0.52\sigma - \mu}{\sigma} = 0.52 $$

  3. Look up z-scores in standard normal distribution tables Using a standard normal distribution table, we find the probabilities corresponding to $z_1 = -0.52$ and $z_2 = 0.52$.

  4. Calculate the percentage difference The area under the normal curve between these two z-scores gives us the percentage of observations that fall within this range. The percentage can be calculated as: $$ P(-0.52 < Z < 0.52) = P(Z < 0.52) - P(Z < -0.52) $$

  5. Use z-table values From standard normal distribution tables:

  • For $z = -0.52$, the area is approximately 0.3015.
  • For $z = 0.52$, the area is approximately 0.7019.

Thus: $$ P(-0.52 < Z < 0.52) = 0.7019 - 0.3015 = 0.4004 $$

  1. Convert to percentage To express this as a percentage, multiply by 100: $$ 0.4004 \times 100 \approx 40.04% $$

Approximately $40.04%$ of observations differ from the mean by less than $0.52\sigma$.

More Information

In a normal distribution, about 68% of data falls within one standard deviation from the mean, demonstrating how we can use z-scores to understand different ranges of deviations from the mean.

Tips

  • Confusing the z-scores: Ensure you calculate the z-scores correctly by using the proper formula.
  • Misreading z-table values: Always verify the correct probability values from the z-table to avoid miscalculations.
  • Failing to convert area to percentage: Remember to multiply the final decimal result by 100 to express it in percentage form.
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