Question

the graph shows the distribution of the cost of drinks at a popular coffee shop. the distribution is approximately normal, with a mean of $3.34 and a standard deviation of $0.75. what percentage of coffee drinks cost within $1.50 of the mean? 34% 68% 95% 99.7%

Explanation:

Step1: Calculate the number of standard deviations

We know that the mean $\mu = 3.34$ and the standard deviation $\sigma=0.75$. We want to find out how many standard deviations $1.50$ is. Let $k$ be the number of standard deviations. Then $k=\frac{1.50}{0.75}$.
$$k = 2$$

Step2: Use the empirical rule for normal distribution

The empirical rule (68 - 95 - 99.7 rule) states that for a normal distribution:

• Approximately $68\%$ of the data lies within $1$ standard deviation ($\mu\pm\sigma$) of the mean.
• Approximately $95\%$ of the data lies within $2$ standard deviations ($\mu\pm2\sigma$) of the mean.
• Approximately $99.7\%$ of the data lies within $3$ standard deviations ($\mu\pm3\sigma$) of the mean.

Since $k = 2$ (from step 1), the percentage of data within $k = 2$ standard deviations of the mean is $95\%$ according to the empirical rule.

Answer:

95%