Questions: The results of a certain medical test are normally distributed with a mean of 120 and a standard deviation of 20. Convert the given results into z-scores, and then use the accompanying table of z-scores and percentiles to find the percentage of people with readings between 154 and 160. The percentage of people with readings between 154 and 160 is % (Round to two decimal places as needed.)

Solution Steps

To convert the test results into \( z \)-scores, we use the formula:

\[ z = \frac{X - \mu}{\sigma} \]

For \( X = 154 \):

\[ z = \frac{154 - 120}{20} = 1.7 \]

For \( X = 160 \):

\[ z = \frac{160 - 120}{20} = 2.0 \]

Thus, the calculated \( z \)-scores are:

• \( z \) for 154: \( 1.7 \)
• \( z \) for 160: \( 2.0 \)

Next, we compute the cumulative distribution function (CDF) values for the \( z \)-scores:

• CDF for \( X = 154 \): \( CDF(154) \approx 0.9554 \)
• CDF for \( X = 160 \): \( CDF(160) \approx 0.9772 \)

To find the percentage of people with readings between 154 and 160, we calculate the difference between the two CDF values:

\[ \text{Percentage} = (CDF(160) - CDF(154)) \times 100 \]

Substituting the values:

\[ \text{Percentage} = (0.9772 - 0.9554) \times 100 \approx 2.18\% \]

Final Answer