Answer:
And we can use the z score given by:
The z score for 3128 is:
And we can use the normal standard table or excel and we got:
For the other probability we have:
And we can use the z score given by:
The z score for 3234 is:
And we can use the complement rule, the normal standard table or excel and we got:
And the final probability would be 0.1606+0.1606= 0.3212
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The central limit theorem states that "if we have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large".
Let X the random variable of interest and we know that:
We select a random sample of n=97 nails. That represent the sample size.
And we want to calculate the probability that the mean weight of the sample babies would differ from the population mean by greater than 53 grams
From the central limit theorem we know that the distribution for the sample mean is given by:
We can find the individual probabilities and we got:
And we can use the z score given by:
The z score for 3128 is:
And we can use the normal standard table or excel and we got:
For the other probability we have:
And we can use the z score given by:
The z score for 3234 is:
And we can use the complement rule, the normal standard table or excel and we got:
And the final probability would be 0.1606+0.1606= 0.3212