Question

the graph shows a distribution of data with a standard deviation of 6. which statement is true about the data point 84? it is within 1 standard deviation of the mean. it is within 2 standard deviations of the mean. it is exactly 3 standard deviations from the mean. it is exactly 4 standard deviations from the mean.

Explanation:

Step1: Identify the mean

The peak of the normal distribution (the dashed line) is at \( x = 75 \), so the mean \( \mu = 75 \). The standard deviation \( \sigma = 6 \).

Step2: Calculate 1, 2, 3, 4 standard deviations from the mean

• 1 standard deviation: \( \mu \pm \sigma = 75 \pm 6 = (69, 81) \)
• 2 standard deviations: \( \mu \pm 2\sigma = 75 \pm 12 = (63, 87) \)
• 3 standard deviations: \( \mu \pm 3\sigma = 75 \pm 18 = (57, 93) \)
• 4 standard deviations: \( \mu \pm 4\sigma = 75 \pm 24 = (51, 99) \)

Step3: Check where 84 lies

The data point 84 is between 63 and 87 (within 2 standard deviations), between 57 and 93 (within 3 standard deviations), but let's check the options:

• "Within 1 standard deviation": 84 > 81 (upper bound of 1σ), so no.
• "Within 2 standard deviations": 84 is between 63 and 87, so yes.
• "Exactly 3 standard deviations": \( 75 + 3(6) = 93 \), 84 ≠ 93 and \( 75 - 3(6) = 57 \), 84 ≠ 57, so no.
• "Exactly 4 standard deviations": \( 75 + 4(6) = 99 \), 84 ≠ 99 and \( 75 - 4(6) = 51 \), 84 ≠ 51, so no.

Answer:

It is within 2 standard deviations of the mean.