Question

The population has mean μ=29 and standard deviation σ=9.

This distribution is shown with the black dotted line.

We are asked for the mean and standard deviation of the sampling distribution for a sample of size 34. The Central Limit Theorem states that the sample mean of a sample of size n is normally distributed with mean μx¯=μ and σx¯=σn√.

In our case, we have μ=29, σ=9, and n=34. So,

μx¯=29

and

σx¯=934‾‾‾√=1.5

This distribution is shown with the red solid line. Notice the sampling distribution, which represents the sample mean of random values of the population, has the same mean as the population distribution. However, the sample mean will vary less than a random value from the population, and therefore has a smaller standard deviation.

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Use the Central Limit Theorem for Means to find the sample mean and the sample standard deviation

Question

What is the probability that the sample mean for a sample of size 34 will be more than 32?

Use the results from above in your calculation and round your answer to the nearest percent. You may use a calculator or the portion of the z-table given below.

z0.40.50.60.70.80.91.01.11.21.31.41.51.61.71.81.92.02.12.22.32.42.52.60.000.65540.69150.72570.75800.78810.81590.84130.86430.88490.90320.91920.93320.94520.95540.96410.97130.97720.98210.98610.98930.99180.99380.99530.010.65910.69500.72910.76110.79100.81860.84380.86650.88690.90490.92070.93450.94630.95640.96490.97190.97780.98260.98640.98960.99200.99400.99550.020.66280.69850.73240.76420.79390.82120.84610.86860.88880.90660.92220.93570.94740.95730.96560.97260.97830.98300.98680.98980.99220.99410.99560.030.66640.70190.73570.76730.79670.82380.84850.87080.89070.90820.92360.93700.94840.95820.96640.97320.97880.98340.98710.99010.99250.99430.99570.040.67000.70540.73890.77040.79950.82640.85080.87290.89250.90990.92510.93820.94950.95910.96710.97380.97930.98380.98750.99040.99270.99450.99590.050.67360.70880.74220.77340.80230.82890.85310.87490.89440.91150.92650.93940.95050.95990.96780.97440.97980.98420.98780.99060.99290.99460.99600.060.67720.71230.74540.77640.80510.83150.85540.87700.89620.91310.92790.94060.95150.96080.96860.97500.98030.98460.98810.99090.99310.99480.99610.070.68080.71570.74860.77940.80780.83400.85770.87900.89800.91470.92920.94180.95250.96160.96930.97560.98080.98500.98840.99110.99320.99490.99620.080.68440.71900.75170.78230.81060.83650.85990.88100.89970.91620.93060.94290.95350.96250.96990.97610.98120.98540.98870.99130.99340.99510.99630.090

Answer 1