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According to a report from a business intelligence company, smartphone owners are using an average of 20 apps per month Assume that number of apps used per month by smartphone owners is normally distributed and that the standard deviation is 4. Complete parts (a) through (d) below. a. If you select a random sample of 36 smartphone owners, what is the probability that the sample mean is between 19.5 and 20.5? (Round to three decimal places as needed.)

Answer :

JeanaShupp

Answer: 0.547

Step-by-step explanation:

As per given , we have

Population mean = [tex]\mu=20[/tex]

Population standard deviation= [tex]\sigma=4[/tex]

Sample size : n= 36

We assume that number of apps used per month by smartphone owners is normally distributed.

Let [tex]\overline{x}[/tex] be the sample mean.

Formula : [tex]z=\dfrac{\overline{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

The probability that the sample mean is between 19.5 and 20.5 :-

[tex]P(19.5<\overline{x}<20.5)\\\\=P(\dfrac{19.5-20}{\dfrac{4}{\sqrt{36}}}<\dfrac{\overline{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}<\dfrac{20.5-20}{\dfrac{4}{\sqrt{36}}})\\\\=P(-0.75<z<0.75)\\\\=P(z<0.75)-P(z<-0.75)\ \ [\because P(z_1<z<z_2)=P(z<z_2)-P(z<z_1)]\\\\=P(z<0.75)-(1-P(z<0.75))\ \ [\because P(Z<-z)=1-P(Z<z)]\\\\=2P(z<0.75)-1=2(0.7734)-1=0.5468\approx0.547[/tex]

[using standard normal distribution table for z]

Hence, the required probability = 0.547

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