Question

To study bonding between mothers and infants, a researcher places each mother and her infant in a playroom and has the mother leave for 10 minutes. The researcher records crying time in the sample of infants during this time that the mother was not present and finds that crying time is normally distributed with M= 7 and SD = 1.1.

Based on the empirical rule, state the range of crying times within 68% of infants cried, 95% of infants cried, and 99.7% of infants cried.

Answer 1

The range of crying times within 68% of the data is (5.9, 8.1).

The range of crying times within 95% of the data is (4.8, 9.2).

The range of crying times within 99.7% of the data is (3.7, 10.3).

Explanation:

According to the Empirical Rule in a normal distribution with mean µ and standard deviation σ, nearly all the data will fall within 3 standard deviations of the mean. The empirical rule can be broken into three parts:

• 68% data falls within 1 standard deviation of the mean. That is P (µ - σ ≤ X ≤ µ + σ) = 0.68.
• 95% data falls within 2 standard deviations of the mean. That is P (µ - 2σ ≤ X ≤ µ + 2σ) = 0.95.
• 99.7% data falls within 3 standard deviations of the mean. That is P (µ - 3σ ≤ X ≤ µ + 3σ) = 0.997.

The mean and standard deviation are:

µ = 7

σ = 1.1

Compute the range of crying times within 68% of the data as follows:

`P(\mu-\sigma\leq X\leq \mu+\sigma)=0.68\\\\P(7-1.1\leq X\leq 7+1.1)=0.68\\\\P(5.9\leq X\leq 8.1)=0.68`

The range of crying times within 68% of the data is (5.9, 8.1).

Compute the range of crying times within 95% of the data as follows:

`P(\mu-2\sigma\leq X\leq \mu+2\sigma)=0.95\\\\P(7-2.2\leq X\leq 7+2.2)=0.95\\\\P(4.8\leq X\leq 9.2)=0.95`

The range of crying times within 95% of the data is (4.8, 9.2).

Compute the range of crying times within 99.7% of the data as follows:

`P(\mu-3\sigma\leq X\leq \mu+3\sigma)=0.997\\\\P(7-3.3\leq X\leq 7+3.3)=0.997\\\\P(3.7\leq X\leq 10.3)=0.997`

The range of crying times within 99.7% of the data is (3.7, 10.3).