Question

In: Statistics and Probability

A random sample of n = 25 observations is taken from a N(µ, σ ) population....

A random sample of n = 25 observations is taken from a N(µ, σ ) population. A 95% confidence interval for µ was calculated to be (42.16, 57.84). The researcher feels that this interval is too wide. You want to reduce the interval to a width at most 12 units.

a) For a confidence level of 95%, calculate the smallest sample size needed.

b) For a sample size fixed at n = 25, calculate the largest confidence level 100(1 − α)% needed.

Solutions

Expert Solution

a) The sample mean is computed as the mid point of the given confidence interval. It is computed as:

From standard normal tables, we have:

P( -1.96 < Z < 1.96 ) = 0.95

Therefore the margin of error here is computed as:

Now for confidence interval width as 12, and above standard deviation the minimum sample size is computed as:

Therefore 43 is the minimum sample size required here.

b) Here for n = 25, we need to find the critical z value first. It is computed as

We now have to find the probability now:

P( -1.5 < Z < 1.5)

= 2*P(0 < Z < 1.5)

From standard normal tables, we have:

P(Z < 1.5) = 0.9332

Therefore P( 0 < Z < 1.5) = 0.9332 - 0.5 = 0.4332

Therefore the required probability here is:

= 2*P(0 < Z < 1.5) = 2*0.4332 = 0.8664

Therefore the largest confidence interval here is given as 86.64%


Related Solutions

A simple random sample of 25 observations will be taken from a population that is assumed...
A simple random sample of 25 observations will be taken from a population that is assumed to be normal with a standard deviation of 37. We would like to test the alternative hypothesis that the standard deviation is actually more than 37. If the sample standard deviation is 40 or more, then the null hypothesis will be rejected (this called a decision rule). What significance level would this decision rule cause? If the population standard deviation is really 42, what...
A random sample of n observations is selected from a population with standard deviation σ =...
A random sample of n observations is selected from a population with standard deviation σ = 1. Calculate the standard error of the mean (SE) for these values of n. (Round your answers to three decimal places.) (a) n = 1 SE = (b) n = 2 SE = (c) n = 4 SE = (d) n = 9 SE = (e) n = 16 SE = (f) n = 25 SE = (g) n = 100 SE =
A simple random sample of size n=49 is obtained from a population with µ=83 and σ=21....
A simple random sample of size n=49 is obtained from a population with µ=83 and σ=21. Describe the sampling distribution of x̄, is this distribution approximately normal. (enter Yes or No) What is P(x̄>88.1)=
Let X ∼ N(µ, σ) and X¯ be sample mean from a random sample of 9....
Let X ∼ N(µ, σ) and X¯ be sample mean from a random sample of 9. Suppose you draw a random sample of 9, calculate an interval ¯x ± 0.5σ where σ is the population standard deviation of X, and then check whether µ, the population mean, is contained in the interval or not. If you repeat this process 100 times, about how many time do you think µ is contained in X¯ ± 0.5σ. Explain why. (Hint: What is...
A random sample of size 40 is taken from a population with mean µ = 240...
A random sample of size 40 is taken from a population with mean µ = 240 and standard deviation σ = 26. i. Describe the probability distribution of the sample mean. ii. What are the mean and the standard deviation of the sample mean? iii. Calculate the probability that the sample mean is between 230 and 250.
A random sample of size 40 is taken from a population with mean µ = 240...
A random sample of size 40 is taken from a population with mean µ = 240 and standard deviation σ = 26. i. Describe the probability distribution of the sample mean. ii. What are the mean and the standard deviation of the sample mean? iii. Calculate the probability that the sample mean is between 230 and 250.
A random sample of size n = 225 is taken from a population with a population...
A random sample of size n = 225 is taken from a population with a population proportion P = 0.55. [You may find it useful to reference the z table.] a. Calculate the expected value and the standard error for the sampling distribution of the sample proportion. (Round "expected value" to 2 decimal places and "standard error" to 4 decimal places.) b. What is the probability that the sample proportion is between 0.50 and 0.60? (Round “z” value to 2...
A random sample of size n = 130 is taken from a population with a population...
A random sample of size n = 130 is taken from a population with a population proportion p = 0.58. (You may find it useful to reference the z table.) a. Calculate the expected value and the standard error for the sampling distribution of the sample proportion. (Round "expected value" to 2 decimal places and "standard error" to 4 decimal places.) b. What is the probability that the sample proportion is between 0.50 and 0.70? (Round “z” value to 2...
5. A random sample of size n = 36 is obtain from a population with µ=...
5. A random sample of size n = 36 is obtain from a population with µ= 60 and standard deviation 12 (a). Describe the sampling distribution . (b). What is P(76.5 << 85.5 )?
A random sample of size n = 36 is obtain from a population with µ= 80,...
A random sample of size n = 36 is obtain from a population with µ= 80, and standard deviation =18. (a). Describe the sampling distribution . (b). What is P(77 << 85 )?
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT