Sampling Distribution

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4.The average breaking strength of a certain steel cable is 2000 pounds, with a standard deviation of 100 pounds. A sample of 20 cables is selected and tested. Find the sample mean that will cut off the upper 95% of all samples of size 20 taken from the population. Assume that the variable is normally distributed.

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To find the sample mean that will cut off the upper 95% of all samples of size 20, we need to find the Z-score associated with the upper 5% tail of the standard normal distribution.

1. Determine the critical value (Z-score) for the upper 5% tail.
- Since the distribution is assumed to be normal, we can use the standard normal distribution table or a calculator.
- The upper 5% tail corresponds to an area of 0.05. Half of this area is allocated to each tail, so the area in the upper tail is 0.05/2 = 0.025.
- Looking up the Z-score associated with an area of 0.025 in the standard normal distribution table or using a calculator will give us the critical value.

2. Calculate the Z-score associated with the upper 5% tail.
- Let's denote the Z-score as Z.
- The Z-score is calculated using the formula: Z = (x - μ) / σ, where x is the sample mean, μ is the population mean, and σ is the standard deviation.
- Since we want to find the sample mean that cuts off the upper 5% tail, we substitute Z = Z_critical into the formula.

3. Rearrange the formula to solve for the sample mean x.
- Rewrite the formula as: x = Z_critical * σ + μ.

Let's now calculate the sample mean.

1. Determine the critical value (Z-score) for the upper 5% tail.
- The upper 5% tail corresponds to an area of 0.025.
- From the standard normal distribution table, the Z-score associated with an area of 0.025 is approximately 1.96.

2. Calculate the Z-score associated with the upper 5% tail.
- Z = Z_critical = 1.96 (approximately)

3. Rearrange the formula to solve for the sample mean x.
- x = Z_critical * σ + μ
= 1.96 * 100 + 2000
= 196 + 2000
= 2196 pounds.

Therefore, the sample mean that will cut off the upper 95% of all samples of size 20 taken from the population is 2196 pounds.