Respuesta :
Answer:
64.76% probability that the mean weight of the sample babies would differ from the population mean by less than 40 grams
Step-by-step explanation:
To solve this question, we have to understand the normal probability distribution and the central limit theorem.
Normal probability distribution:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central limit theorem:
The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex]
In this problem, we have that:
[tex]\mu = 3242, \sigma = 446, n = 107, s = \frac{446}{\sqrt{107}} = 43.12[/tex]
What is the probability that the mean weight of the sample babies would differ from the population mean by less than 40 grams
This is the pvalue of Z when X = 3242 + 40 = 3282 subtracted by the pvalue of Z when 3242 - 40 = 3202
X = 3282
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{3282 - 3242}{43.12}[/tex]
[tex]Z = 0.93[/tex]
[tex]Z = 0.93[/tex] has a pvalue of 0.8238.
X = 3202
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{3202 - 3242}{43.12}[/tex]
[tex]Z = -0.93[/tex]
[tex]Z = -0.93[/tex] has a pvalue of 0.1762
0.8238 - 0.1762 = 0.6476
64.76% probability that the mean weight of the sample babies would differ from the population mean by less than 40 grams
