Answer:
1. 15.87%
2. 6 pounds and 8.8 pounds.
3. 2.28%
4. 50% of newborn babies weigh more than 7.4 pounds.
5. 84%
Step-by-step explanation:
We are given the following information in the question:
Mean, μ = 7.4 pounds
Standard Deviation, σ = 0.7 pounds
We are given that the distribution of weights for newborn babies is a bell shaped distribution that is a normal distribution.
Formula:
1.Percent of newborn babies weigh more than 8.1 pounds
P(x > 8.1)
Calculation the value from standard normal z table, we have,
15.87% of newborn babies weigh more than 8.1 pounds.
2.The middle 95% of newborn babies weight
Empirical Formula:
- Almost all the data lies within three standard deviation from the mean for a normally distributed data.
- About 68% of data lies within one standard deviation from the mean.
- About 95% of data lies within two standard deviations of the mean.
- About 99.7% of data lies within three standard deviation of the mean.
Thus, from empirical formula 95% of newborn babies will lie between
![\mu-2\sigma= 7.4-2(0.7) = 6\\\mu+2\sigma= 7.4+2(0.7)=8.8](https://tex.z-dn.net/?f=%5Cmu-2%5Csigma%3D%207.4-2%280.7%29%20%3D%206%5C%5C%5Cmu%2B2%5Csigma%3D%207.4%2B2%280.7%29%3D8.8)
95% of newborn babies will lie between 6 pounds and 8.8 pounds.
3. Percent of newborn babies weigh less than 6 pounds
P(x < 6)
Calculation the value from standard normal z table, we have,
2.28% of newborn babies weigh less than 6 pounds.
4. 50% of newborn babies weigh more than pounds.
The normal distribution is symmetrical about mean. That is the mean value divide the data in exactly two parts.
Thus, approximately 50% of newborn babies weigh more than 7.4 pounds.
5. Percent of newborn babies weigh between 6.7 and 9.5 pounds
![P(6.7 \leq x \leq 9.5)\\\\ = P(\displaystyle\frac{6.7 - 7.4}{0.7} \leq z \leq \displaystyle\frac{9.5-7.4}{0.7})\\\\ = P(-1 \leq z \leq 3)\\\\= P(z \leq 3) - P(z < -1)\\= 0.9987 -0.1587= 0.84 = 84\%](https://tex.z-dn.net/?f=P%286.7%20%5Cleq%20x%20%5Cleq%209.5%29%5C%5C%5C%5C%20%3D%20P%28%5Cdisplaystyle%5Cfrac%7B6.7%20-%207.4%7D%7B0.7%7D%20%5Cleq%20z%20%5Cleq%20%5Cdisplaystyle%5Cfrac%7B9.5-7.4%7D%7B0.7%7D%29%5C%5C%5C%5C%20%3D%20P%28-1%20%5Cleq%20z%20%5Cleq%203%29%5C%5C%5C%5C%3D%20P%28z%20%5Cleq%203%29%20-%20P%28z%20%3C%20-1%29%5C%5C%3D%200.9987%20-0.1587%3D%200.84%20%3D%2084%5C%25)
84% of newborn babies weigh between 6.7 and 9.5 pounds.