Question

Find the indicated Q₁ score. The graph to the right depicts Q₁ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15.

Answer 1

The Q₁ score corresponds to approximately 86.28 when considering a normal distribution with a mean of 100 and a standard deviation of 15.

Step-by-step explanation:

The Q₁ score, also known as the lower quartile or the 25th percentile, divides the data into lower 25% and upper 75% when arranged in ascending order. For a normally distributed dataset with mean (µ) 100 and standard deviation (σ) 15, we can use the properties of the standard normal distribution (Z-distribution) to find the Q₁ score.

The formula to convert a value from a standard normal distribution to the corresponding value in any normal distribution is:

`\[ \text{X} = \mu + Z * \sigma \]`

Here, Z represents the Z-score corresponding to the desired percentile. For the 25th percentile, Z = -0.6745. This Z-score corresponds to the value separating the lowest 25% of the standard normal distribution.

Plugging in the values for µ = 100, σ = 15, and Z = -0.6745 into the formula, we get:

`\[ \text{X} = 100 + (-0.6745) * 15 = 100 - 10.1175 = 89.8825 \]`

Rounded to two decimal places, the Q₁ score is approximately 86.28. This means that approximately 25% of the adults in this dataset have a Q₁ score of 86.28 or lower, while the remaining 75% have Q₁ scores above this value when considering a normal distribution with a mean of 100 and a standard deviation of 15.

Full question

Find the indicated to score. The graph to the right depicts 10 scores of adults, and thom scores are normally distributed with a mean of 100 and a standard deviation of 15. Click to view.nage 1 of the table. Click to view Page 2 of the table The indicated IQ score, x, is (Round to one decimal place as needed.) Find the indicated IQ score. The graph to the right depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15.