Question

Describe examples of situations where you could appropriately use the following. Explain why the procedure is the correct one to use and identify the statistic to be used and explain why that is the correct choice.

a. Confidence interval for a proportion.

b. Confidence interval for a mean, unknown σ.

c. Confidence interval for a mean, σ known.

Describe procedure for following and provide an example of the calculations:

d. Margin of error calculation for 95% confidence level.

e. Sample size determination for 95% confidence level

Answer 1

Describe examples of situations where you could appropriately use the following. Explain why the procedure is the correct one to use and identify the statistic to be used and explain why that is the correct choice.

a. Confidence interval for a proportion.

We can use confidence interval for a proportion for finding the interval for proportion or percentage of the people who voted for particular political party. By using this confidence interval, we can get the range of percentages of people who will vote for particular political party. There are several situations where we can use this confidence interval. Where we need to compute the interval for the percentages of the particular characteristic, we can use this interval. Let us consider another example. Suppose, researcher want to find the approximate range of percentages of products produced in shifts which are defected, then researcher can use this confidence interval for population proportion. For finding the approximate range of percentages of the people who have cars, we can use this confidence interval. The confidence interval for the population proportion is given as below:

Confidence Interval = P ± Z* sqrt(P*(1 – P)/n)

Where, P is the sample proportion, Z is critical value, and n is sample size.

b. Confidence interval for a mean, unknown σ.

When population standard deviation σ is unknown, then we use the t confidence interval for the population mean. There are many situations we need to find the confidence interval for the population mean, but we don’t know the population standard deviation. For example, suppose we want to find the confidence interval for the population mean for the average expenditure of the person in the particular city. Also, we want to find the confidence interval for the average age of the players who represent their national team. The confidence interval formula is given as below:

Confidence interval = Xbar ± t*S/sqrt(n)

c. Confidence interval for a mean, σ known.

When population standard deviation σ is known, then we use the z confidence interval for the population mean. There are many situations we need to find the confidence interval for the population mean, where we know the population standard deviation. For example, suppose we want to find the confidence interval for the population mean for the daily data usage for Smartphone users in the city, given the population standard deviation of the data usage. Also, we want to find the confidence interval for the average age of the players who represent their national team, where we know the population standard deviation for the age of the players from the previous data. The confidence interval formula is given as below:

Confidence interval = Xbar ± Z*σ/sqrt(n)

d. Margin of error calculation for 95% confidence level.

The formula for margin of error is given as below:

Margin of error = E = Critical value*Standard error

Critical values and standard errors will be different for different types of confidence interval.

Suppose we want to find the margin of error for Confidence interval for a mean, where σ is known.

We are given 95% confidence level, so critical Z value = 1.96

Suppose, population standard deviation = σ = 12

Sample size = n = 36

Standard error = σ/sqrt(n) = 12/sqrt(36) = 12/6 = 2

Margin of error = E = Critical value*Standard error

Margin of error = E = 1.96*2

Margin of error = E = 3.92

e. Sample size determination for 95% confidence level

Suppose, we are given

Population standard deviation = σ = 12

Margin of error = E = 2

Confidence level = 95%

Critical Z value = 1.96

Sample size formula is given as below:

n = (Z*σ/E)^2

n = (1.96*12/2)^2

n = 138.2976

n = 139

Required sample size = 139