# Let x the mean of a sample of size 25 since μ x 90 σ

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LetX¯= the mean of a sample of size 25. Sinceμx= 90,σx= 15, andn= 25,X¯~N90,1525.FindP85 <x¯< 92. Draw a graph.P85 <x¯< 92= 0.6997The probability that the sample mean is between 85 and 92 is 0.6997.Figure 7.2normalcdf(lower value, upper value, mean, standard error of the mean)The parameter list is abbreviated (lower value, upper value,μ,σn)normalcdf(85,92,90,1525) = 0.6997366Chapter 7 | The Central Limit TheoremThis OpenStax book is available for free at
b. Find the value of the sample mean that is two standard deviations above the expected value, 90.Solution 7.1b. To find the value that is two standard deviations above the expected value 90, use thez-score formula:zx¯=x¯μxσxnwhere the number of standard deviations,zx¯, is 2, the expected value,μx, is 90, the standarddeviation of the original distribution,σxis 15, and the sample sizenis 25.Plugging in the known values and solving forx¯2 =x¯− 9015/252 =x¯− 9036 =x¯− 90x¯= 96The value of the sample mean that is two standard deviations above the expected value is 96.The standard error of the mean isσxn=1525= 3. Recall that the standard error of the mean is a description ofhow far (on average) that the sample mean will be from the population mean in repeated simple random samplesof sizen.7.1An unknown distribution has a mean of 45 and a standard deviation of eight. Samples of sizen= 30 are drawnrandomly from the population. Find the probability that the sample mean is between 42 and 50.Example 7.2The length of time, in hours, it takes an "over 40" group of people to play one soccer match is normally distributedwith amean of two hoursand astandard deviation of 0.5 hours. Asample of sizen= 50is drawn randomlyfrom the population. Find the probability that thesample meanis between 1.8 hours and 2.3 hours.Solution 7.2LetX= the time, in hours, it takes to play one soccer match.The probability question asks you to find a probability for thesample mean time, in hours, it takes to play onesoccer match.LetX¯= themeantime, in hours, it takes to play one soccer match.Ifμx= _________,σx= __________, andn= ___________, thenX~N(______, ______) by thecentrallimit theorem for means.μx= 2,σx= 0.5,n= 50, andX~N2,0.550Chapter 7 | The Central Limit Theorem367
FindP1.8 <x¯< 2.3. Draw a graph.P1.8 <x¯< 2.3= 0.9977normalcdf1.8,2.3,2,.550= 0.9977The probability that the mean time is between 1.8 hours and 2.3 hours is 0.9977.7.2The length of time taken on the SAT for a group of students is normally distributed with a mean of 2.5 hours and astandard deviation of 0.25 hours. A sample size ofn= 60 is drawn randomly from the population. Find the probabilitythat the sample mean is between two hours and three hours.To find percentiles for means on the calculator, follow these steps.2ndDIStR3:invNormk= invNormarea to the left ofk, mean,standarddeviationsamplesizewhere:k= thekthpercentilemeanis the mean of the original distributionstandarddeviationis the standard deviation of the original distributionsamplesize=nExample 7.3In a recent study reported Oct. 29, 2012 on the Flurry Blog, the mean age of tablet users is 34 years. Suppose thestandard deviation is 15 years. Take a sample of sizen= 100.

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