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calculate the probability that the sample mean?
A manufacturer of power tools claims that the average amount of time required to assemble their top-of-the-line table saw is fifty (50) minutes with a standard deviation of forty (40) minutes (the very large standard deviation is due to a variety of factors including a large variation in skills amongst the ‘Do it yourself’ home handyman which is traditionally one of the companies customer base). Suppose a random sample of 64 purchasers of this table saw is taken.
What is the probability that the sample mean will be more than 44 minutes __________?
The Central Limit Theorem tells us that for any sample, if the sample size is sufficently large, then the mean will follow the normal distribution with a mean equal to the sample mean and a variance equal to the variance of the sample divided by the sample size. In other words, if we have a sample with mean μ and variance σ² then the sample mean, Xbar, will follow the normal distribution with mean μ and variance σ²/n or a standard deviation of σ/√(n). Note: if you do not have the population parameters υ or σthen approximate them with the sample values of xbar and s.
In this question we have
Xbar ~ Normal( μ = 50 , σ² = 1600 / 64 )
Xbar ~ Normal( μ = 50 , σ² = 25 )
Xbar ~ Normal( μ = 50 , σ = 40 / sqrt( 64 ) )
Xbar ~ Normal( μ = 50 , σ = 5 )
Find P( Xbar > 44 )
P( ( Xbar - μ ) / σ > ( 44 - 50 ) / 5 )
= P( Z > -1.2 )
= P( Z < 1.2 )
= 0.8849303
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