Question

when andrew goes bowling, his scores are normally distributed with a mean of 160 and a standard deviation of 13. using the empirical rule, what percentage of the games that andrew bowls does he score between 134 and 186?

Explanation:

Step1: Find how many standard deviations 134 and 186 are from the mean.

The mean ($\mu$) is 160 and the standard deviation ($\sigma$) is 13.
For 134: $160 - 134 = 26$, and $\frac{26}{13}=2$. So 134 is 2 standard deviations below the mean ($\mu - 2\sigma$).
For 186: $186 - 160 = 26$, and $\frac{26}{13}=2$. So 186 is 2 standard deviations above the mean ($\mu + 2\sigma$).

Step2: Apply the empirical rule.

The empirical rule states that for a normal distribution:

• Approximately 68% of data lies within $\mu \pm \sigma$.
• Approximately 95% of data lies within $\mu \pm 2\sigma$.
• Approximately 99.7% of data lies within $\mu \pm 3\sigma$.

Since we found that 134 is $\mu - 2\sigma$ and 186 is $\mu + 2\sigma$, the percentage of data between them is approximately 95%.

Answer:

95%