One-Sample t-test
Remember In the last chapter, we learned to use a statistic from a large sample of data to test a hypothesis about a population parameter. In our case, using a z-test, we tested a hypothesis about a population mean using the mean of one sample drawn from that population. We agreed that by "large sample" we mean any sample that has 30 or more data values.
Something to think about What happens, however, when we have a sample with less than 30 items?
Answer: Here, we cannot use the one-sample z-test. But what we will is its counterpart, the onesample t-test.
The History of t-test
William Gossett He is a chemist, who worked for Guinness in the early 1900s.
William Gossett His job involved testing samples of ale to ensure quality.
William Gossett He was determined to develop a statistical tool that would allow him to use smaller and still be able to test hypothesis about a population parameter.
The t Distribution Gossett, while experimenting with possible solutions, noticed something interesting about the mound-shaped distribution of data values when the sample size was less than 30.
The t Distribution Each time he decreased the sample size by one, and plotted the means of repeated samples of the same size, the shape of the distribution flattened out.
The t Distribution
The t Distribution From the figure, we can conclude that the fewer the number of data values you have, the more spread out on both ends (i.e., platykurtosis) our distribution is.
The t Distribution Gossett further discovered that the empirical rule applied to this distribution and, if you compensate for the number of data values less than 30, you can test a hypothesis by comparing a computed value of t to a critical value of t.
The One-Sample t-test This is a statistical tool use when testing a hypothesis of a sample that has less than 30 members/items.
Example 13.1. "Persons living in urban environments have anxiety levels significantly different from the national average."
Example 13.1. Let's say we have a dataset with 15 values to test this hypothesis and the national average anxiety level is 30. Also, we have a test that is used to measure anxiety and the scores from it range from 0 (no anxiety) to 80 (high anxiety).
Data
Determining the Computed Value of t
Formula for Computed Value of t
Determining the Critical Value of t
The Area Under the Normal Curve Table We will determine our critical value of t using a table developed especially for that purpose the table showing the area under the curve for t values.
Degrees of Freedom (df) This is the number of values in the dataset which are collected that are "free to vary" when trying to estimate any given value. We will denote this using minuscule letters df.
Understanding the Degrees of Freedom (df) Suppose we have a one-sample t-test where we know the population mean is 10. I might ask you to tell me five numbers that, when summed, would give an average of 10. You might start out by saying "6, 8, 10, 12,..." but then I would have to interrupt you by saying, "That is enough."
To sum it up If we know the mean and all of the values in the dataset except one, we can easily determine the missing value. In this case, four of our values in the dataset can vary, but the fifth one cannot.
Formula for Computing the Degrees of Freedom (df):
Question: Using the formula for finding df, what is the df in Example 13.1?
Danger in Computing Degrees of Freedom
Danger: The degrees of freedom is computed in different ways. It will depend on the number of items in your dataset and the number of levels of the independent variable.
Danger: The degrees of freedom is computed in different ways. It will depend on the number of items in your dataset and the number of levels of the independent variable.
Challenge: Compare the Critical Values of t Table to that of Areas Under the Normal Curve Table.
Using Critical Values of t Table
Example 13.1.1. In order to use the t table, let us suppose we have a scenario with 12 degrees of freedom and want the critical value for alpha.05. What is the critical value of t?
Example 13.1.2. We might have 29 degrees of freedom and a critical value of t when alpha is.025. What us the critical value of t?
Going Back to Example 13.1. "Persons living in urban environments have anxiety levels significantly different from the national average."
Question: What have you noticed about the type of hypothesis we have? Then, what do you think should we do with our alpha value? So, what is now our alpha value and what is now our critical value?
Task: Plotting Our Critical Value of t
Analyzing the Result By comparing our computed value of t and critical value of t and looking at the figure, what should be our decision?
Decision: Since our computed value of t (i.e., 2.691) is not within the range of the critical values, we can reject the null hypothesis. By looking at our data results, it appears that the anxiety levels of people living in urban areas may be significantly higher than the national average.
Testing Hypothesis Using the p Value and Alpha Value
Table 13.3. Inferential Statistics from a One-Sample Test
Question: How do we compute for the p value for nondirectional hypothesis?
Decision: We can see based on the Table 13.3 that our p value, labeled "Sig. (2- tailed)" is.018; this is less than our alpha value of.025. This verifies our decision we made when we compared the critical value of t to the computed value of t; we must reject our null hypothesis and support our research hypothesis.
Something to think about Although we have rejected the null hypothesis, some statisticians feel this isn't enough. At this point, we know the groups being compared are significantly different, but we know nothing about the degree to which they're different, right?
Effect Size Indices This is a statistical tool developed to help us better understand the magnitude of any significant difference which uncovered. It is also called as practical significance and Cohen's delta.
Practical Significance/ Cohen s Delta It helps us better understand the extent to which the independent variable affects the dependent variable.
Formula for Computing the Effect Size:
Danger in Computing the Effect Size Standard Error of the Mean is different from Standard Deviation of the Mean.
Question: Using the formula, what is our Cohen's delta?
Answer: This leaves us with an effect size of.695, but what does it mean?
Computed Effect Size It is the percentage of the standard deviation that the difference in the mean scores represents.
Cohen s Effect Size Interpretations
Small If the computed Cohen's delta is.2 or smaller.
Medium If the computed Cohen's delta is between.2 and.5.
Large If the computed Cohen's delta is greater than.5.
Therefore, The computed effect size of.695 is large effect size; the groups are significantly different, and the levels of the independent variable had a dramatic effect on the dependent variable. In short, there is a strong relationship between where the persons live and their level of anxiety.
Example 13.2. "Men who use anabolic steroids will have significantly shorter life expectancies than men who do not."
Example 13.2. In this case, suppose we know the average life expectancy for men is 70 years but we only have access to information on 12 men who used steroids of this type.
Example 13.2.
Task 1: Compute for the Computed Value of t using the formula
Task 2: Determine the alpha value.
Question: Do we need to divide our alpha value by 2? Why or why not?
Task 3: Plot our critical value on the "less than" side of the distribution.
Question: Based on the critical value of t and computed value of t, and by looking at our distribution, what decision can you make?
Decision: Our computed t value, - 1.852, is less than the critical value of t (i.e., -1.796). Because of that, we reject our null hypothesis and support our research hypothesis.
Task 4: Compute for the value of Cohen's delta.
Note: We are interested only in the absolute value, so we drop the negative sign and wind up with an effect size if.535.
Decision: Our previous decision is supported by a large effect size of.535; the independent variable does have a large effect on the dependent variable. Apparently, men who use anabolic steroids live significantly fewer years than their peers who do not.
Danger in Using the p Value
Note 1: If we are dealing with a nondirectional hypothesis (twotailed hypothesis) and if we divide the alpha value by 2, then we need to use the entire p value. By dividing our alpha value, we already allowed an equal probability of error on both sides of the distribution.
Note 2: If we are dealing with directional hypothesis (one-tailed hypothesis) and if we used the entire alpha value, then we need to divide our p value by 2. This is for the fact that we are just looking at one side or one direction of our hypothesis, which is either less than or greater than.
Danger in Using the p Value
Deepening Our Knowledge about One-Sample t-test Now that we know how to test a hypothesis by both comparing a computed value of t to a critical value of t and comparing a computed p value to a pre established alpha value, let's put this all together and use our sixstep model.
The Six-Step Statistical Model
Step 1: Identify the Problem
Characteristics of a Good Problem Statement The problem is interesting to the researcher.
Characteristics of a Good Problem Statement The scope of the problem is manageable by the researcher.
Characteristics of a Good Problem Statement The researcher has the knowledge, time, and resources needed to investigate the problem.
Characteristics of a Good Problem Statement The problem can be researched through the collection and analysis of numeric data.
Characteristics of a Good Problem Statement Investigating the problem has theoretical or practical significance.
Characteristics of a Good Problem Statement It is ethical to investigate the problem.
Step 2: State a Hypothesis
Step 3: Identify the Independent Variable
Step 4: Identify and Describe the Dependent Variable
Step 5: Choose the Right Statistical Test
Step 6: Data Computation and Analysis to Test the Hypothesis
Case 1: Residents in a lower socioeconomic section of town recently complained to local health officials that ambulance response times to their neighborhood were not as fast as average response times throughout town. This, of course, worried the officials, and they decided to investigate. Their records showed the average response time throughout the city to be 90 seconds after an emergency was phoned in. They decided to monitor the next 20 response times to the concerned neighborhood to help determine if the complaint had any merit.
Step 1: Identify the Problem
Question: What do you think is the problem which we need to investigate?
Answer: Yes, the city officials are interested in this problem.
Question: Will this meet our criteria for a good research problem?
Answer: Yes, the scope of the problem is manageable by city officials. They would simply need to collect response times for each section of town.
Question: Does the researcher possess knowledge, time, and resources needed to investigate the problem?
Answer: Yes, the city officials have the knowledge, time, and resources needed to investigate the problem.
Question: Is the problem can be researched through collection and analysis of numeric data?
Answer: Yes, the problem can be researched through the collection and analysis of numeric data.
Statement of the Problem: "This study will investigate whether ambulance response times vary between different sections of the city."
Step 2: State the Hypothesis How will we state our null hypothesis?
Null Hypothesis "Ambulance response times to the lower socioeconomic community are not significantly greater than 90 seconds."
Step 3: Identify the Independent Variable
Question: What is our independent variable? Why?
Answer: In this case we are interested in looking at times for 20 ambulance calls to the affected neighborhood (i.e., our sample) and compare them to a known population parameter (i.e., the average time for the total number of calls in the town). Because of that, our independent variable is "calls for an ambulance."
Question: How many levels do we have in our independent variable?
Answer: We only have one level; calls to the specific part of the town. While it seems that calls to the town overall are another variable, that is not the case. They represent the overall population and cannot be considered as a level of an independent variable.
Step 4: Identify and Describe the Dependent Variable
Question: What is our dependent variable?
Answer: The dependent variable is the average response time for the 20 ambulance response times in the sample.
Data:
Step 5: Choosing the Right Statistical Tool How many independent variable do we have? How many level? How many dependent variable?
Answer: We have one independent variable with one level. We also have one dependent variable representing quantitative data. As a result, we are going to use the one-sample t-test.
Step 6: Data Computation and Analysis to Test the Hypothesis
Task 1 Compute the computed Value of t
Task 2 Determine the Critical Value of t
Task 3 Construct and plot the values on the normal distribution
Task 4 Determine the p value
Task 5 Compare the p value to alpha value
Task 6 Determine the effect size
Task 7 Make a decision and conclusion
Decision and Conclusion: Our one-tailed p value is much smaller than our alpha value (i.e.,.05), so we reject the null hypothesis. This significant difference is supported by a somewhat large effect size of.520. This is further substantiated by noting that our computed value of t, 2.326, is much larger than our critical value of t, 1.729
Let s Practice
Direction: Read and analyze each case below. Answer it by using the Six-Step Statistical Model.
The Case of Stopping Sneezing Physicians at a premiere hospital are constantly working to lower the amount of time it takes a patient to stop sneezing after being exposed to an allergen. Their experience has shown, on average that it takes about 2 minutes for standard drugs to be effective. Today, however, they are listening to a representative from a pharmaceutical company who is introducing a drug the company claims will end sneezing in significantly less time. The physicians skeptically agreed to try the drug and decided to use it on a sample of ten patients. The results of that sample, they believe, will help them decide whether to use the new "miracle drug."
The Case of Stopping Sneezing
The Case of Stopping Sneezing I have lived in big cities most of my life and recently decided to move out into the country. One of the pleasures I've discovered is the ability to grow your own garden and "live off the fruit of the land." In talking to some of the other folks around here, I've discovered that many people consider growing tomatoes a science somewhat akin to nuclear physics. They talk about the amount of rain needed, the type of fertilizer to use, how far plants should be spaced apart, and other really exciting topics. One thing I found interesting is that they consider a 10-ounce tomato to be about average; most folks can't grow them any bigger than that. After hearing that, I figured a good way for a city boy to fit in would be to grow the biggest, juiciest tomatoes in town! In order to set my plan in action, I went to the local "feed and seed" store and inquired as to which fertilizer I should use. The guy running the place said he had basically two types. The cheaper variety worked well enough, but he told me, by spending a few extra pesos, I could grow tomatoes I would probably be proud of! Given that, I bought the costlier of the two, drove home, and put my plan into action!
The Case of Stopping Sneezing