The following appeared in the magazine Financial Times, March 23,1995: "When Elvis Presley died in 1977, there were 48 professio
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Answer:
a)
b)
c)
Step-by-step explanation:
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 number of apps used per month by smartphone users. We know from the problem that the distribution for the random variable X is given by:
We take a sample of n=25 . That represent the sample size.
From the central limit theorem we know that the distribution for the sample mean is also normal and is given by:
a. what is the probability that the sample mean is between 29 and 31?
In order to answer this question we can use the z score in order to find the probabilities, the formula given by:
And we want to find this probability:
b. what is the probability that the sample mean is between 28 and 32?
c. If you select a random sample of 100 smartphone owners, what is the probability that the sample mean is between 29 and 31?
On this case we have a new distribution for the sample mean given by:
And we want to find this probability:
On this case the results for parts a and b shows the probability that we have values within one and two deviations from the mean. And the result for part c its equal for part b since the new deviation for the sampel mean with a sample size of 100 its the half of the standard deviation when we use a random sample of 25.