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Wize University Statistics Textbook > Estimating with Confidence Intervals

Margin of Error and Sample Size

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Margin of Error and Sample Size

Recall that the margin of error is:
z∗p^q^n\boxed{\displaystyle{z^{\ast}\sqrt{ \frac{\hat{p}\hat{q}}{n}}}}

Suppose we have a desired margin of error, mm. Given the confidence level, we can find the sample size, n⋆n^\star, to give us the desired margin of error.

Example
If p^=0.4\hat{p}=0.4 and we want a 95% confidence interval with a margin of error of 0.05. What should be our sample size?

Since m=z∗p^q^nm=z^{\ast}\sqrt{ \frac{\hat{p}\hat{q}}{n}}

we use algebra to isolate n and get
n∗=(z∗)2p^q^m2n^{\ast}=\frac{\left(z^{\ast}\right)^2\hat{p}\hat{q}}{m^2}
z∗=1.96z^{\ast}=1.96 (for 95% confidence)

Solve:

n∗=(1.96)2(0.4)(1−0.4)(0.05)2\displaystyle{}n^{\ast}=\frac{\left(1.96\right)^2\left(0.4\right)\left(1-0.4\right)}{\left(0.05\right)^2}

n∗=368.79 ≈369n^{\ast}=368.79\ \approx369 (rounded up!)


Wize Tip
Always round UP to the next whole number!

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How do we reduce the margin of error?

z∗p^q^n\boxed{\displaystyle{z^{\ast}\sqrt{ \frac{\hat{p}\hat{q}}{n}}}}
  1. Increase n; or
  2. Decrease z⋆z^\star (opt for a lower confidence interval)
Suppose the sample proportion is 60% and we want to construct a 95% confidence interval with a margin of error of 3%.
Extra Practice
Margin of Error and Sample Size
Approximately 70% of laptops in our sample have components that are made in China. How many laptops should you sample to estimate the true proportion with 95% confidence and a margin of error of 0.075?
Margin of Error and Sample Size
In a sample random sample of 20, the average is 18 and the standard deviation is 1.25. Let m1m_1 be the margin of error for a 95% confidence interval. Let m2m_2 be the margin of error for a 90% confidence interval. Calculate the value of m1−m2m_1-m_2
Margin of Error and Sample Size
Previous studies have shown that approximately 70% of laptops have components that
are made in Cina. How many laptops should you sample to estimate the true proportion with 95% confidence and a margin of error of 0.09?
Margin of Error and Sample Size
You wish to estimate the proportion of all laptops in Kanata that have components made in Cina. In a random sample of 12 laptops, 10 have components made in Cina.
The margin of error for a 95% confidence interval for the proportion of laptops made in Cina is
Margin of error and sample size
Professor Karl has two large statistics classes. He teaches one class in the morning and one in the evening. He samples students in both classes to assess his teaching performance. He uses their latest quiz marks for his assessment.
Here is some summarized statistics:
Assuming that the variances of the class quiz marks are equal, find the margin of error for a 95% confidence interval for the difference in the two means
Margin of Error and Sample Size
In a sample random sample of 20, the average is 18 and the standard deviation is 1.25. Let m1m_1 be the margin of error for a 95% confidence interval. Let m2m_2 be the margin of error for a 90% confidence interval. Calculate the value of m1−m2m_1-m_2
Margin of Error and Sample Size
A research group wants to estimate the number of transactions people make on their credit cards each month. The standard deviation is known to be 42 transactions. They want to construct a 95% confidence interval to estimate the mean number of transactions for October 2015 to within no more than 8 transactions.
Determine the sample size required.
What sample size is required if they want to use a 99% confidence interval?
Margin og Error and Sample Size
A statistician wishes to compute a 90% confidence interval for the average number of calories a typical adult consumes. He wishes to have a margin of error of no more than 10 calories. What is the minimum sample size he can use? The population standard deviation is 75.
Margin of Error
Increasing the sample size will decrease the margin of error while keeping the same level of confidence.
Margin of Error and Sample Size
A research group wants to estimate the number of transactions people make on their credit cards each month. The standard deviation is known to be 42 transactions. They want to construct a 95% confidence interval to estimate the mean number of transactions for October 2015 to within no more than 8 transactions.
Margin of Error and Sample Size
A statistician wishes to compute a 90% confidence interval for the average number of calories a typical adult consumes. He wishes to have a margin of error of no more than 10 calories. What is the minimum sample size he can use? The population standard deviation is 75.
Margin of Error and Sample Size
True or False?
1) Increasing n will increase the margin of error?
2) Increasing Zα2Z_{\frac{α}{2}} will increase the margin of error?
Margin of Error and Sample Size
In predicting the mean length of a species of snakes, how many times as many observations do we need to sample to cut the margin of error by a half without changing the level of confidence?
Confidence Interval for a Mean
Which statement is true about a 95% confidence interval [188,212]?
Statistical errors
For the following statements, select true or false. Enter T for True, F for False. (Use capital letters: T, F)
1. The mean weight of five cats is 20 lbs. We take away one that at weights 28 lbs. The mean weight of the remaining cats is 18 lbs.
T
2. The 95% confidence interval is [$800, $1,200]. The sample mean must be $1,000.
T