Question

birth weights of babies born to full - term pregnancies follow a normal distribution with a mean weight of 7.2 pounds with a standard deviation of 1.1 pound. without using z - score(s), what percent of babies born to full - time pregnancies weigh between 4.5 and 5.5 pounds? (all problems in this edla were done using the formulas and the \by hand\ method. if you are using a calculator then round your answers to 3 decimal places and you should be able to match the correct answer.)
05.41%
68.00%
90.91%
94.59%

Explanation:

Step1: Calculate the number of standard deviations from the mean

The mean $\mu = 7.2$ and the standard deviation $\sigma=1.1$.
For $x = 4.5$: $7.2-4.5 = 2.7$, and $2.7\div1.1\approx2.45$ (more than 2 standard deviations below the mean).
For $x = 5.5$: $7.2 - 5.5=1.7$, and $1.7\div1.1\approx1.55$ (more than 1 standard deviation below the mean).
We know that the total area under the normal curve is 100%. The area within 1 standard deviation of the mean is about 68%, within 2 standard deviations is about 95%, and within 3 standard deviations is about 99.7%.
The area below 1.55 standard deviations below the mean:
The area below the mean is 50%. The area between the mean and 1.55 standard deviations below the mean: using the property of the normal distribution, we can calculate.
The area below 2.45 standard deviations below the mean:
We know that the area within 2 standard deviations of the mean is about 95%, so the area outside 2 standard deviations is $100 - 95=5\%$, and the area below 2 standard deviations below the mean is $5\%\div2 = 2.5\%$.
The area within 1.55 standard deviations of the mean:
The area within 1 standard deviation is 68%, and we can use the symmetry and properties of the normal distribution (using the formula for the normal distribution $P(aUsing the formula $P(4.5From the standard - normal table (or using the property that $\varPhi(-z)=1-\varPhi(z)$), $\varPhi(-1.55)=1 - \varPhi(1.55)\approx1 - 0.9394 = 0.0606$ and $\varPhi(-2.45)=1-\varPhi(2.45)\approx1 - 0.9929=0.0071$
$P(4.5 < X < 5.5)=0.0606-0.0071 = 0.0535\approx5.41\%$

Answer:

05.41%