If the sample size increases, the margin of error of a confidence interval will
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Remember that there is variability associated with your outcomes and statistics. Show
When you calculate a statistic based on your sample data, how do you know if the statistic truly represents your population? Even if you've selected a random sample, your sample will not completely reflect your population. Each sample you take will give you a different result. Let's Look at an Example:Suppose that you want to compare the mean age for those with and without an IV in the prehospital setting. You review the ambulance runs for the past two weeks and calculate a mean age of 10.4 years for those with an IV and 8.5 years for those without an IV. The difference between the two means is 1.9 years. From this, you might conclude that those receiving an IV were older on average. Suddenly, it's not clear that there's an important difference in age between these two groups. Now suppose you collect the same data over the next six weeks. This time the average age for those with an IV is 9.2 years and the average age for those without an IV is 8.9 years, for a difference of 0.3 years. Suddenly, it's not clear that there's an important difference in age between these two groups. Why did your different samples yield different results? Is one sample more correct than the other? Remember that there is variability in your outcomes and statistics. The more individual variation you see in your outcome, the less confidence you have in your statistics. In addition, the smaller your sample size, the less comfortable you can be asserting that the statistics you calculate are representative of your population. Providing a Range of ValuesA confidence interval provides a range of values that will capture the true population value a certain percentage of the time. You determine the level of confidence, but it is generally set at 90%, 95%, or 99%. Confidence intervals use the variability of your data to assess the precision or accuracy of your estimated statistics. You can use confidence intervals to describe a single group or to compare two groups. We will not cover the statistical equations for a confidence interval here, but we will discuss several examples. Example The confidence interval in the above example could be described at 94 to 108 bpm (beats per minute) or 101 bpm ± 7 bpm. Here the number 7 is your margin of error. For confidence intervals around the mean, the margin of error is just half of your total confidence interval width. Sample Size and VariabilityThe precision of your statistics depends on your sample size and variability. A larger sample size or lower variability will result in a tighter confidence interval with a smaller margin of error. A smaller sample size or a higher variability will result in a wider confidence interval with a larger margin of error. The level of confidence also affects the interval width. If you want a higher level of confidence, that interval will not be as tight. A tight interval at 95% or higher confidence is ideal. Examples:  rev. 04-Aug-2022 IntroductionWhile you are learning statistics, you will often have to focus on a sample rather than the entire population. This is because it is extremely costly, difficult and time-consuming to study the entire population. The best you can do is to take a random sample from the population – a sample that is a ‘true’ representative of it. You then carry out some analysis using the sample and make inferences about the population. Since the inferences are made about the population by studying the sample taken, the results cannot be entirely accurate. The degree of accuracy depends on the sample taken – how the sample was selected, what the sample size is, and other concerns. Common sense would say that if you increase the sample size, the chances of error will be less because you are taking a greater proportion of the population. A larger sample is likely to be a closer representative of the population than a smaller one. Let’s consider an example. Suppose you want to study the scores obtained in an examination by students in your college. It may be time-consuming for you to study the entire population, i.e. all students in your college. Hence, you take out a sample of, say, 100 students and find out the average scores of those 100 students. This is the sample mean. Now, when you use this sample mean to infer about the population mean, you won’t be able to get the exact population means. There will be some “margin of error”. You will now learn the answers to some important questions: What is margin of error, what are the method of calculating margins of error, how do you find the critical value, and how to decide on t-score vs z-scores. Thereafter, you’ll be given some margin of error practice problems to make the concepts clearer. What is Margin of Error?The margin of error can best be described as the range of values on both sides (above and below) the sample statistic. For example, if the sample average scores of students are 80 and you make a statement that the average scores of students are 80 ± 5, then here 5 is the margin of error. Calculating Margins of ErrorFor calculating margins of error, you need to know the critical value and sample standard error. This is because it’s calculated using those two pieces of information. The formula goes like this: margin of error = critical value * sample standard error. How do you find the critical value, and how to calculate the sample standard error? Below, we’ll discuss how to get these two important values. How do You find the Critical Value?For finding critical value, you need to know the distribution and the confidence level. For example, suppose you are looking at the sampling distribution of the means. Here are some guidelines.
It is important to know the distribution to decide what to use – t-scores vs z-scores. Caution – when your sample size is large and it is not given that the distribution is normal, then by Central Limit Theorem, you can say that the distribution is normal and use z-score. However, when the sample size is small and it is not given that the distribution is normal, then you cannot conclude anything about the normality of the distribution and neither z-score nor t-score can be used. When finding the critical value, confidence level will be given to you. If you are creating a 90% confidence interval, then confidence level is 90%, for 95% confidence interval, the confidence level is 95%, and so on. Here are the steps for finding critical value: Step 1: First, find alpha (the level of significance). \alpha =1 – Confidence level. For 95% confidence level, \alpha =0.05 For 99% confidence level, \alpha =0.01 Step 2: Find the critical probability p*. Critical probability will depend on whether we are creating a one-sided confidence interval or a two-sided confidence interval. For two-sided confidence interval, p*=1-\dfrac { \alpha }{ 2 } For one-sided confidence interval, p*=1-\alpha Then you need to decide on using t-scores vs z-scores. Find a z-score having a cumulative probability of p*. For a t-statistic, find a t-score having a cumulative probability of p* and the calculated degrees of freedom. This will be the critical value. To find these critical values, you should use a calculator or respective statistical tables. Sample Standard ErrorSample standard error can be calculated using population standard deviation or sample standard deviation (if population standard deviation is not known). For sampling distribution of means: Let sample standard deviation be denoted by s, population standard deviation is denoted by \sigma and sample size be denoted by n. \text {Sample standard error}=\dfrac { \sigma }{ \sqrt { n } }, if \sigma is known \text {Sample standard error}=\dfrac { s }{ \sqrt { n } }, if \sigma is not known Depending on the sampling distributions, the sample standard error can be different. Having looked at everything that is required to create the margin of error, you can now directly calculate a margin of error using the formula we showed you earlier: Margin of error = critical value * sample standard error. Some Relationships1. Confidence level and marginal of errorAs the confidence level increases, the critical value increases and hence the margin of error increases. This is intuitive; the price paid for higher confidence level is that the margin of errors increases. If this was not so, and if higher confidence level meant lower margin of errors, nobody would choose a lower confidence level. There are always trade-offs! 2. Sample standard deviation and margin of errorSample standard deviation talks about the variability in the sample. The more variability in the sample, the higher the chances of error, the greater the sample standard error and margin of error. 3. Sample size and margin of errorThis was discussed in the Introduction section. It is intuitive that a greater sample size will be a closer representative of the population than a smaller sample size. Hence, the larger the sample size, the smaller the sample standard error and therefore the smaller the margin of error. Margin of Error Practice ProblemsExample 1 25 students in their final year were selected at random from a high school for a survey. Among the survey participants, it was found that the average GPA (Grade Point Average) was 2.9 and the standard deviation of GPA was 0.5. What is the margin of error, assuming 95% confidence level? Give correct interpretation. Step 1: Identify the sample statistic. Since you need to find the confidence interval for the population mean, the sample statistic is the sample mean which is the average GPA = 2.9. Step 2: Identify the distribution – t, z, etc. – and find the critical value based on whether you need a one-sided confidence interval or a two-sided confidence interval. Since population standard deviation is not known and the sample size is small, use a t distribution. \text {Degrees of freedom}=n-1=25-1=24. \alpha=1-\text {Confidence level}=1-0.95=0.05 Let the critical probability be p*. For two-sided confidence interval, p*=1-\dfrac { \alpha }{ 2 } =1-\dfrac { 0.05 }{ 2 } =0.975. The critical t value for cumulative probability of 0.975 and 24 degrees of freedom is 2.064. Step 3: Find the sample standard error. \text{Sample standard error}=\dfrac { s }{ \sqrt { n } } =\dfrac { 0.5 }{ \sqrt { 25 } } =0.1 Step 4: Find margin of error using the formula: Margin of error = critical value * sample standard error = 2.064 * 0.1 = 0.2064 Interpretation: For a 95% confidence level, the average GPA is going to be 0.2064 points above and below the sample average GPA of 2.9. Example 2 400 students in Princeton University are randomly selected for a survey which is aimed at finding out the average time students spend in the library in a day. Among the survey participants, it was found that the average time spent in the university library was 45 minutes and the standard deviation was 10 minutes. Assuming 99% confidence level, find the margin of error and give the correct interpretation of it. Step 1: Identify the sample statistic. Since you need to find the confidence interval for the population mean, the sample statistic is the sample mean which is the mean time spent in the university library = 45 minutes. Step 2: Identify the distribution – t, z, etc. and find the critical value based on whether the need is a one-sided confidence interval or a two-sided confidence interval. The population standard deviation is not known, but the sample size is large. Therefore, use a z (standard normal) distribution. \alpha=1-\text{Confidence level}=1-0.99=0.01 Let the critical probability be p*. For two-sided confidence interval, p*=1-\dfrac { \alpha }{ 2 } =1-\dfrac { 0.01 }{ 2 } =0.995. The critical z value for cumulative probability of 0.995 (as found from the z tables) is 2.576. Step 3: Find the sample standard error. \text{Sample standard error}=\dfrac { s }{ \sqrt { n } } =\dfrac { 10 }{ \sqrt { 400 } } =0.5 Step 4: Find margin of error using the formula: Margin of error = critical value * sample standard error = 2.576 * 0.5 = 1.288 Interpretation: For a 99% confidence level, the mean time spent in the library is going to be 1.288 minutes above and below the sample mean time spent in the library of 45 minutes. Example 3 Consider a similar set up in Example 1 with slight changes. You randomly select X students in their final year from a high school for a survey. Among the survey participants, it was found that the average GPA (Grade Point Average) was 3.1 and the standard deviation of GPA was 0.7. What should be the value of X (in other words, how many students you should select for the survey) if you want the margin of error to be at most 0.1? Assume 95% confidence level and normal distribution. Step 1: Find the critical value. \alpha=1-\text{Confidence level}=1-0.95=0.05 Let the critical probability be p*. For two-sided confidence interval, p*=1-\dfrac { \alpha }{ 2 } =1-\dfrac { 0.05 }{ 2 } =0.975. The critical z value for cumulative probability of 0.975 is 1.96. Step 3: Find the sample standard error in terms of X. \text{Sample standard error}=\dfrac { s }{ \sqrt { X } }=\dfrac { 0.7 }{ \sqrt { X } } Step 4: Find X using margin of error formula: Margin of error = critical value * sample standard error 0.1=1.96*\dfrac { 0.7 }{ \sqrt { X } } This gives X=188.24. Thus, a sample of 189 students should be taken so that the margin of error is at most 0.1. ConclusionThe margin of error is an extremely important concept in statistics. This is because it is difficult to study the entire population and the sampling is not free from sampling errors. The margin of error is used to create confidence intervals, and most of the time the results are reported in the form of a confidence interval for a population parameter rather than just a single value. In this article, you made a beginning by learning answering questions like what is margin of error, what is the method of calculating margins of errors, and how to interpret these calculations. You also learned to decide whether to use t-scores vs z-scores and gained information about finding critical values. Now you know how to use margin of error for constructing confidence intervals, which are widely used in statistics and econometrics. Let’s put everything into practice. Try this Statistics practice question:Looking for more Statistics practice?You can find thousands of practice questions on Albert.io. Albert.io lets you customize your learning experience to target practice where you need the most help. We’ll give you challenging practice questions to help you achieve mastery in Statistics. Start practicing here. Are you a teacher or administrator interested in boosting Statistics student outcomes? Learn more about our school licenses here. What happens to margin of error when sample size increases?The margin of error decreases as the sample size increases because the Law of Large Numbers states that as the sample size increases the sample mean approaches the value of the population mean.
What happens to the margin of error if the confidence interval increases?Increasing the confidence will decrease the margin of error resulting in a narrower interval.
What happens to the margin of error as the sample size decreases?The larger the sample size, the smaller the margin of error. Conversely, the smaller the sample size, the larger the margin of error.
Why does increasing sample size reduce margin of error?It is intuitive that a greater sample size will be a closer representative of the population than a smaller sample size. Hence, the larger the sample size, the smaller the sample standard error and therefore the smaller the margin of error.
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