Based on the normal population mean and the standard deviation, the probability that a sample $57 and $64 is 0.5761.
The probability that the mean is between $57 and $64 can be shown as:
P (57 < x < 64) = P ( ( lower limit - Upper limit) / Standard deviation / √ sample mean) < Z < P ( ( lower limit - Upper limit) / Standard deviation / √ sample mean)
Solving gives:
P (57 < x < 64) = P ( ( 57 - 61 ) / 13 / √ 9) < Z < ( ( 64 - 61 ) / 13 / √ 9)
= P( -0.92 < Z < 0.69)
Using the Z-table:
= P( Z < 0.69) - (P (Z < -0.92)
= 0.7549 - 0.1788
= 0.5761
In conclusion, the probability that a sample mean is between $64 and $57 is 0.5761.
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