Questions: A researcher who is in charge of an educational study wants subjects to perform some special skill. Fearing that people who are unusually talented or unusually untalented could distort the results, he decides to use people who scored in the middle 50% on a certain test. If the mean for the population is 100 and the standard deviation is 10, find the two limits (upper and lower) for the scores that would enable a volunteer to participate in the study. Assume the scores are normally distributed. Refer to the table of values (*) Area Under the Standard Normal Distribution) as needed. If necessary, round your answer to the nearest whole number. A volunteer with a test score between and can participate in the study.

A researcher who is in charge of an educational study wants subjects to perform some special skill. Fearing that people who are unusually talented or unusually untalented could distort the results, he decides to use people who scored in the middle 50% on a certain test. If the mean for the population is 100 and the standard deviation is 10, find the two limits (upper and lower) for the scores that would enable a volunteer to participate in the study. Assume the scores are normally distributed.

Refer to the table of values (*) Area Under the Standard Normal Distribution) as needed. If necessary, round your answer to the nearest whole number.

A volunteer with a test score between and can participate in the study.

Transcript text: A researcher who is in charge of an educational study wants subjects to perform some special skill. Fearing that people who are unusually talented or unusually untalented could distort the results, he decides to use people who scored in the middle $50 \%$ on a certain test. If the mean for the population is 100 and the standard deviation is 10, find the two limits (upper and lower) for the scores that would enable a volunteer to participate in the study. Assume the scores are normally distributed Refer to the table of values (*) Area Under the Standard Normal Distribution) as needed. If necessary, round your answer to the nearest whole number. A volunteer with a test score between $\square$ and $\square$ can participate in the study.

Solution

Solution Steps

Step 1: Define the Problem

The researcher aims to include subjects who scored in the middle $50\%$ of a normally distributed test score population. The population has a mean ($\mu$) of $100$ and a standard deviation ($\sigma$) of $10$.

Step 2: Identify the Percentiles

To find the limits for the middle $50\%$ of scores, we need to determine the $25^{th}$ percentile ($P_{25}$) and the $75^{th}$ percentile ($P_{75}$). These percentiles correspond to the z-scores:

$z_{25} \approx -0.6745$
$z_{75} \approx 0.6745$

Step 3: Calculate the Limits

Using the z-score formula: \[ X = \mu + z \cdot \sigma \] we can calculate the lower and upper limits:

For the lower limit ($L$): \[ L = 100 + (-0.6745) \cdot 10 = 100 - 6.745 = 93.255 \approx 93 \]
For the upper limit ($U$): \[ U = 100 + (0.6745) \cdot 10 = 100 + 6.745 = 106.745 \approx 107 \]

Step 4: Conclusion

A volunteer with a test score between $93$ and $107$ can participate in the study.