Mathematics, 08.04.2020 01:35 Anybody6153

In a study reported in the Flurry Blog on Oct. 29, 2012, the mean age of tablet users was 34 yrs, with a standard deviation of 15 years. Assuming a normal distribution, what is the approximate probability of picking a random sample of 40 tablet users with a mean age between 31 and 35 yrs?

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Answer from: laskew37221

55.90% probability of picking a random sample of 40 tablet users with a mean age between 31 and 35 yrs

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 $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

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 $\mu$ and standard deviation $\sigma$ , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$ .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem, we have that:

$\mu = 34, \sigma = 15, n = 40, s = \frac{15}{\sqrt{40}} = 2.37$

Assuming a normal distribution, what is the approximate probability of picking a random sample of 40 tablet users with a mean age between 31 and 35 yrs?

This is the pvalue of Z when X = 35 subtracted by the pvalue of Z when X = 31.

X = 35

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{35 - 34}{2.37}$

Z = 0.42

Z = 0.42 has a pvalue of 0.6628

X = 31

$Z = \frac{X - \mu}{s}$

$Z = \frac{31 - 34}{2.37}$

Z = -1.26

Z = -1.26 has a pvalue of 0.1038

0.6628 - 0.1038 = 0.5590

55.90% probability of picking a random sample of 40 tablet users with a mean age between 31 and 35 yrs

Answer from: Quest

Here, feel free to ask if there are any questions

Answer from: Quest

$9.89

step-by-step explanation:

9.90 = p x .06 x .25 (3/12 changed to a decimal)

9.90 = p x 0.015

9.90 - 0.015 = p

9.89 = p

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