The Tite Wire Company manufactures wires for circus acts. It has taken a
random sample of 100 pieces of wire and wants to see if the thickness of a batch
of wire meets minimum specifications. Assume that the population mean is
0.45 inch, with a standard deviation of 0.03 inch.
a) Calculate the mean and the standard deviation of the sampling distribution.
b) What may be said about the shape of the sampling distribution?
c) Within what range of values does the sample mean have a 68.26% chance of
falling?
d) Within what range of values does the sample mean have a 95.44% chance of
falling?
a. 0.45
b. 0.003
a) To calculate the mean and standard deviation of the sampling distribution, you can use the following formulas:
Mean of the sampling distribution (μ): It is equal to the population mean.
μ = 0.45 inch
Standard deviation of the sampling distribution (σ): It is equal to the population standard deviation divided by the square root of the sample size.
σ = population standard deviation / √sample size
= 0.03 inch / √100
= 0.03 inch / 10
= 0.003 inch
Therefore, the mean of the sampling distribution is 0.45 inch, and the standard deviation of the sampling distribution is 0.003 inch.
b) The shape of the sampling distribution can be assumed to be approximately normally distributed. This is based on the Central Limit Theorem, which states that if the sample size is sufficiently large (usually greater than 30), then the sampling distribution of the mean will be approximately normally distributed, regardless of the shape of the population distribution.
c) The range in which the sample mean has a 68.26% chance of falling is within one standard deviation from the mean of the sampling distribution.
Lower Limit = μ - σ = 0.45 inch - 0.003 inch = 0.447 inch
Upper Limit = μ + σ = 0.45 inch + 0.003 inch = 0.453 inch
Therefore, the sample mean has a 68.26% chance of falling between 0.447 inch and 0.453 inch.
d) The range in which the sample mean has a 95.44% chance of falling is within two standard deviations from the mean of the sampling distribution.
Lower Limit = μ - 2σ = 0.45 inch - 2 * 0.003 inch = 0.444 inch
Upper Limit = μ + 2σ = 0.45 inch + 2 * 0.003 inch = 0.456 inch
Therefore, the sample mean has a 95.44% chance of falling between 0.444 inch and 0.456 inch.