Population:
Mean (μ): 3500 g
Standard deviation (σ): 500g
(a)
The z-score for this distribution can be calculated using the formula:
[tex]Z=\frac{X-\mu}{\sigma}...(1)[/tex]
The z-score of X = 3100 g is:
[tex]Z=\frac{3100-3500}{500}=\frac{400}{500}=-0.8[/tex]
Now, to find the probability that a baby is born with a weight less than 3100 g, we calculate the probability of Z < -0.8 in a standardized distribution:
[tex]P(X\lt3100\text{ g})=P(Z\lt-0.8)=0.2118554[/tex]
As we can see, the probability is approximately 0.212, and the corresponding shaded area is:
(b)
Sample size = 25
The z-score for the distribution of the sample mean can be calculated using the formula:
[tex]Z=\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}[/tex]
Where n is the sample size. Now, we calculate the z-score for a mean of 3100 g:
[tex]Z=\frac{3100-3500}{500/\sqrt{25}}=\frac{-400}{500/5}=-4[/tex]
Finally, the probability that the 25 babies are born with a mean weight less than 3100 g is:
[tex]P(Z\lt-4)=0.0000317[/tex]
As we can see, the probability is approximately 0.0000317, and the corresponding shaded area is:
