A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 219.7-cm and a standard dev
2 answers:
Answer:
30.51% probability that the average length of a randomly selected bundle of steel rods is between 219.7-cm and 220-cm.
Step-by-step explanation:
To solve this question, we need 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 and standard deviation
, the zscore of a measure X is given by:
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 normally distributed random variable X, with mean and standard deviation
, the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
In this problem, we have that:
Find the probability that the average length of a randomly selected bundle of steel rods is between 219.7-cm and 220-cm.
This is the pvalue of Z when X = 220 subtracted by the pvalue of Z when X = 219.7.
X = 220
By the Central Limit Theorem
has a pvalue of 0.8051
X = 219.7
has a pvalue of 0.5
0.8051 - 0.5 = 0.3051
30.51% probability that the average length of a randomly selected bundle of steel rods is between 219.7-cm and 220-cm.
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Answer:
c. The number of 7's in a randomly selected set of five random digits from a table of random digits.
True, for this case we have a value fixed for n =5 and the probability is defined for each number 1/10 assuming numbers (0,1,2,3,4,5,6,7,8,9) so then the random variable "The number of 7's in a randomly selected set of five random digits" can be modelled with the binomial probability function.
Step-by-step explanation:
Previous concepts
The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".
Solution to the problem
Let X the random variable of interest, on this case we now that:
The probability mass function for the Binomial distribution is given as:
Where (nCx) means combinatory and it's given by this formula:
The conditions to apply this distribution is that we have the parameters fixed n and p.
Let's analyze one by one the possible solutions:
a. The number of traffic tickets written by each police officer in a large city during one month.
False, the number of traffic tickets written by each police is not a fixed amount always, so then the value of n change and we can't apply a binomial model for this case.
b. The number of hearts in a hand of five cards dealt from a standard deck of 52 cards that has been thoroughly shuffled.
False, not all the hands of size 5 are equal and since we can't ensure this condition then the binomial model not apply for this case
c. The number of 7's in a randomly selected set of five random digits from a table of random digits.
True, for this case we have a value fixed for n =5 and the probability is defined for each number 1/10 so then the random variable "The number of 7's in a randomly selected set of five random digits" can be modelled with the binomial probability function.
d. The number of phone calls received in a one-hour period.
False, the number of phone calls change by the hour and is not always fixed so then we don't have a valu for n, and the binomial model not applies for this case.
e. All of the above.
False option C is correct.