A sample of 148 values is randomly selected from a population with mean, μ, equal to 46 and standard deviation, σ, equal to 17. (Give your answers correct to one decimal place.)
(a) Determine the interval (smallest value to largest value) within which you would expect 99.7% of such sample means to lie.
Incorrect: Your answer is incorrect. to Incorrect: Your answer is incorrect.
(b) What is the amount of deviation from the mean for a sample mean of 46.7?
Incorrect: Your answer is incorrect.
(c) What is the maximum deviation you have allowed for in your answer to part (a)?
Incorrect: Your answer is incorrect.
http://davidmlane.com/hyperstat/z_table.html
When μ and σ are known, the sample mean (x̄) has normal distribution so the statistic is Z=(x̄-μ)/(σ/sqrt(n)), where n=sample size=148.
(a)
Assuming a symmetrical confidence interval, we have α=0.003/2=0.0015. Looking up a normal probability table, for example:
http://www.stat.ufl.edu/~athienit/Tables/Ztable.pdf
we find |Z|=2.96774.
and the confidence interval is given by
μ-Zσ/sqrt(n) ≤ μ ≤ μ+Zσ/sqrt(n)
Substitute σ, μ, and Z to find the confidence interval. Round to one place after the decimal.
(b) deviation = |observed value - actual value |.
(c) The deviation allowed in part (a) is the term that we add/subtract from the mean, namely Zσ/sqrt(n).
(a) To determine the interval within which you would expect 99.7% of sample means to lie, we can use the concept of the standard deviation, also known as the standard error of the mean.
The standard error of the mean (SE) can be calculated using the formula:
SE = σ / sqrt(n)
Where σ is the population standard deviation and n is the sample size.
In this case, the population standard deviation σ is given as 17, and the sample size n is 148.
SE = 17 / sqrt(148)
SE ≈ 1.39
Based on the Empirical Rule (also known as the 68-95-99.7 rule), 99.7% of the sample means will lie within ±3 standard errors of the mean.
Therefore, the interval within which you would expect 99.7% of sample means to lie is:
46 ± (3 * 1.39) or 46 ± 4.17
This gives us the interval from the smallest value of 46 - 4.17 = 41.83 to the largest value of 46 + 4.17 = 50.17.
So, the interval within which you would expect 99.7% of sample means to lie is approximately 41.8 to 50.2.
(b) To find the amount of deviation from the mean for a sample mean of 46.7, we can calculate the z-score using the formula:
z = (x - μ) / (σ / sqrt(n))
Where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
In this case, x = 46.7, μ = 46, σ = 17, and n = 148.
z = (46.7 - 46) / (17 / sqrt(148))
z ≈ 0.232
To find the amount of deviation, we can multiply the z-score by the standard error (SE):
Deviation = z * SE
Deviation = 0.232 * 1.39
Deviation ≈ 0.323
Therefore, the amount of deviation from the mean for a sample mean of 46.7 is approximately 0.323.
(c) The maximum deviation allowed for in the answer to part (a) is the margin of error, which is the half-width of the interval.
Half-width = 4.17 / 2
Half-width = 2.085
Therefore, the maximum deviation allowed for in the answer to part (a) is approximately 2.085.