Respuesta :
Answer:
37.07% probability that a randmoly chosen penny weighs less than 2.49 grams
Step-by-step explanation:
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.
In this problem, we have that:
[tex]\mu = 2.5, \sigma = 0.03[/tex]
What is the probability that a randmoly chosen penny weighs less than 2.49 grams
This is the pvalue of Z when X = 2.49. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{2.49 - 2.5}{0.03}[/tex]
[tex]Z = -0.33[/tex]
[tex]Z = -0.33[/tex] has a pvalue of 0.3707
37.07% probability that a randmoly chosen penny weighs less than 2.49 grams
Answer:
Probability that a randomly chosen penny weighs less than 2.49 grams is 0.48803.
Step-by-step explanation:
We are given that the weights of United States pennies are distributed approximately normal with a mean of 2.5 grams and a standard deviation of 0.03 grams.
Let X = weights of United States pennies
So, X ~ N([tex]\mu=2.5,\sigma^{2} = 0.03^{2}[/tex])
The z score probability distribution is given by;
Z = [tex]\frac{X-\mu}{\sigma}[/tex] ~ N(0,1)
where, [tex]\mu[/tex] = population mean = 2.5 grams
[tex]\sigma[/tex] = standard deviation = 0.03 grams
So, probability that a randomly chosen penny weighs less than 2.49 grams is given by = P(X < 2.49 grams)
P(X < 2.49) = P( [tex]\frac{X-\mu}{\sigma}[/tex] < [tex]\frac{2.49 -2.5}{0.03}[/tex] ) = P(Z < -0.33) = 1 - P(Z [tex]\leq[/tex] 0.03)
= 1 - 0.51197 = 0.48803
Therefore, probability that a randomly chosen penny weighs less than 2.49 grams is 0.48803.
