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One-sample t-interval for the mean

Page history last edited by GanoHasanbegovic 5 years, 7 months ago

The one-sample t-interval for the mean is a method of statistical inference for quantitative data that that uses an interval derived from a sample to approximate a true population mean. The data of a sample is used to find an interval that will (hopefully) contain the true population mean somewhere within, and give a certain amount of confidence that the interval generated from the sample does contain the true population mean within. This is similar for finding a one-proportion  z-interval for categorical data, but replaces z scores with the Student's t, which helps compensate for extra variability, and is especially useful for data with a small sample size. A one-sample t-interval for the mean is a useful tool for inference by itself, but is also good to use after running a hypothesis test like the one-sample t-test for the mean. The equation for finding the One-sample t-interval for the mean is

 

Formula±t* ×SE(Formula)

 

where

 

SE(Formula)= s/Formula

 

 

When doing a sample t-interval for the mean, it is important to check several conditions. First make sure the sample deals with quantitative data and not categorical data. Next check for independence by checking to see if it is random or at least still representative (since many things cannot truly be random but still represent the population being sampled) and that the sample is less than 10% of the population. Watch out for bias in the sampling method. Finally, make sure the sample meets the nearly normal condition with a histogram and QQ plot. The sample should look unimodal and symmetric. Beware of multimodal severely skewed data, and outliers. However, if the sample size is greater than 40, these should not be an issue, but it is still good to check. Only after examining data for all these things can we then go on to do a one-sample t-interval for the mean. 

 

Suppose we have data about the calorie content of vanilla yogurt from 11 different different brands, and want to use this data to figure out what the real mean value of calories vanilla yogurts have. First we have to check the data to see if it can be used. Now we don't exactly know that these 11 yogurts were chosen randomly, but there is nothing that stands out as a substantial bias so we can safely assume that it is representative. There are probably more than 110 yogurt brands in the world, so our 11 brands are less than 10%. Let's look at the histogram:


The sample is small and the histogram is bulky so let's look at the QQ plot: 

 

 

According to the QQ plot, the data looks nearly normal enough, so we can go on. We decide that we want an interval of 95% confidence to capture the true mean of calories in vanilla yogurt. We can either use the equation above to plug in the values we get from the data ourselves, or use technology to find the intervals for us. If we are doing the work by hand, it is useful to find the critical value tfor 95% confidence and 10 degrees of freedom with the help of technology, or use a table of critical t values of different degrees of freedom and confidence to find the value. For this data we find that  

 

Formula = 131.82, = 25.226, n=11, and t= 2.228.

 

After calculation we find that the interval is (114.87,148.77). This data tells us that 95% of the time, this interval will capture the true population mean, so we can say that "we are 95% confident that the true mean of calories in vanilla yogurts is contained within the interval of 114.87 and 148.77 calories".

 

We can use SPSS to help us with one-sample t-intervals in several ways. Using Analyze-Descriptive Statistics-Explore, we can check the nearly normal condition, find the intervals, and the sample mean and standard deviation. We can also use an IFD.T command to find the critical t value. For the IDF.T command we need to know the degrees of freedom, which will be n-1, and a confidence probability value in decimals. For example, if your amount of confidence is 95%, like with our yogurt example, your intervals will be at the top 97.5% and bottom 2.5%. SPSS shades to the left, so you want the bottom 2.5. In decimal form that is .025, and that is the value you will input into SPSS. SPSS will give you a negative value for the lower interval because it is to the left of 0, so just use the absolute value.  It is important to note that while doing a one-sample t-test for the mean on SPSS, the data will also give you something called "confidence interval of the difference". This output is misleading. Those numbers are not the correct ones, but if you add those numbers to your "test value", they will give you the intervals.

 

 

Generating a one-sample T-Interval in SPSS 

  • Go to the "Analyze" menu.
  • Find the "Descriptive Statistics" submenu and choose the "Explore"command.  
  • Drag the variable you want to analyze into the "Dependent List".  
  • then click on "Plots" and select the "Histogram" and the "Normality plots with test" options.
  • (You may deselect "Stem-and-Leaf" plots, because we don't need it and it can be misleading) Click Continue.  
  • Click on "Statistics". 
  • Make sure the "Descriptives" option is selected, and input the amount of  confidence you want into "Confidence Interval for the Mean".  
  • Click continue, then click OK.
  • Find the "Descriptives" table: the second row contains the confidence intervals, and you can also find the sample mean, and standard deviation if you wanted to compute the intervals by hand.  
  • You can scroll down to the histogram and QQ plots to check the nearly normal condition.  

 

The following video illustrates these steps:

 

Unable to display content. Adobe Flash is required.  

 

Generating a Critical T-Score in SPSS

  • First make sure your data set has been activated by having some sort of data in at least one cell. It can be anything. 
  • Go to the "Transform" menu 
  • Click on "Compute Variable"
  • Put something like "T" in the "target variable" box. (what you write in does not actually matter, but it helps to write in what you're looking for).
  • Make sure the "numeric expression" box is empty, or just click "reset". 
  • Under the "function group" box, scroll down to "Inverse DF" and click on it.
  • Find IDF.T and double click it.  (or in the "numeric expression box write in IDF.T(?,?) )
  • Plug in the confidence probability value for the first "?" and the degrees of freedom for the second "?" (These will take some calculation).
  • Click OK.
  • The value will show up under a new variable name (that you wrote in) in the data set. Your critical t-score will be the absolute value of that number. 

 

The following video illustrates these steps:

 

Unable to display content. Adobe Flash is required.

 

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