Hypothesis Testing > Left Tailed Test or Right Tailed Test ?
Contents (click to skip to that section):
- Hypothesis Testing Basics: One Tail or Two?
- Left Tailed Test or Right Tailed Test? Example
- How to Run a Right Tailed Test
Hypothesis Testing Basics: One Tail or Two?
Watch the video or read the article below:
In a hypothesis test, you have to decide if a claim is true or not. Before you can figure out if you have a left tailed test or right tailed test, you have to make sure you have a single tail to begin with. A tail in hypothesis testing refers to the tail at either end of a distribution curve.
Area under a normal distribution curve. Two tails (both left and right) are shaded.
Basic Hypothesis Testing Steps
- Decide if you have a one-tailed test or a two-tailed test (How to decide if a hypothesis test is a one-tailed test or a two-tailed test). If you have a two-tailed test, you don’t need to worry about whether it’s a left tailed or right tailed test (because it’s both!).
- Find out if it’s a left tailed test or right tailed test (see below).
If you can sketch a graph, you can figure out which tail is in your test.
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Left Tailed Test or Right Tailed Test? Example
Sample question: You are testing the hypothesis that the drop out rate is more than 75% (>75%). Is this a left-tailed test or a right-tailed test?
Step 1: Write your null hypothesis statement and your alternate hypothesis statement. This step is key to drawing the right graph, so if you aren’t sure about writing a hypothesis statement, see: How to State the Null Hypothesis.
Step 2: Draw a normal distribution curve.
Step 3: Shade in the related area under the normal distribution curve. The area under a curve represents 100%, so shade the area accordingly. The number line goes from left to right, so the first 25% is on the left and the 75% mark would be at the left tail.
The yellow area in this picture illustrates the area greater than 75%. From this diagram you can clearly see that it is a right-tailed test, because the shaded area is on the right.
Note: This next picture represent the phrase “less than 25%”. You can see that it would be a left-tailed test from the picture, as the tail is shaded on the left.
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How to Run a Right Tailed Test
Hypothesis tests can be three different types:
The right tailed test and the left tailed test are examples of one-tailed tests. They are called “one tailed” tests because the rejection region (the area where you would reject the null hypothesis) is only in one tail. The two tailed test is called a two tailed test because the rejection region can be in either tail.
Here’s what the right tailed test looks like on a graph:
As you can see, the rejection region (shaded in yellow) is to the right of the graph.
What is a Right Tailed Test?
A right tailed test (sometimes called an upper test) is where your hypothesis statement contains a greater than (>) symbol. In other words, the inequality points to the right. For example, you might be comparing the life of batteries before and after a manufacturing change. If you want to know if the battery life is greater than the original (let’s say 90 hours), your hypothesis statements might be:
Null hypothesis: No change (H0 = 90).
Alternate hypothesis: Battery life has increased (H1) > 90.
The important factor here is that the alternate hypothesis(H1) determines if you have a right tailed test, not the null hypothesis.
Right Tailed Test Example.
A high-end computer manufacturer sets the retail cost of their computers based in the manufacturing cost, which is $1800. However, the company thinks there are hidden costs and that the average cost to manufacture the computers is actually much more. The company randomly selects 40 computers from its facilities and finds that the mean cost to produce a computer is $1950 with a standard deviation of $500. Run a hypothesis test to see if this thought is true.
Step 1: Write your hypothesis statement (see: How to state the null hypothesis).
H0: μ ≤ 1800
H1: μ > 1800
Step 2: Find the test statistic using the z-score formula:
z = 1950 – 1800 / (500/√40) = 1.897
Step 3: Choose an alpha level. No alpha is mentioned in the question, so use the standard (0.05).
1 – 0.05 = .95
Look up that value (.95) in the middle of the z-table. The area corresponds to a z-value of 1.645. That means you would reject the null hypothesis if your test statistic is greater than 1.645.*
1.897 is greater than 1.645, so you can reject the null hypothesis.
* Not sure how I got 1.645? The left hand half of the curve is 50%, so you look up 45% in the “right of the mean” table on this site (50% + 45% = 95%).
A z of 1.645, found by locating .45 in the center of the table (which is actually between two numbers, .4495 and .4505.
This z-table shows the area to the right of the mean, so you’re actually looking up .45, not .95. That’s because half of the area (.5) is not actually showing on the table.
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A hypothesis should be tested to determine if the source has been correctly identified. Investigators use many methods to test their hypotheses. Two main methods are analytic epidemiologic studies and food testing.
Case-control studies or cohort studies are the most common type of analytic study conducted so investigators can analyze information collected from ill persons and comparable well persons to see whether ill persons are more likely than people who did not get sick to have eaten a certain food or to report a particular exposure. Controls for a case-control study may be matched on geography to ensure that cases or ill persons and controls or well persons had the same opportunities for exposure to a contaminated food item. One method to geographically match is to use a reverse digit dialing protocol. A case address is entered into an online directory (such as www.whitepages.com), then a reverse address search is conducted to identify phone numbers for neighbors in that geographic area. Duplicate phone numbers and businesses are excluded.
If eating a particular food is reported more often by sick people than by well people, it may be associated with illness. Using statistical tests, the investigators can determine the strength of the association (i.e., how likely it is to have occurred by chance alone), and whether more than one food might be involved. Investigators look at many factors when interpreting results from these studies:
- Frequencies of exposure to a specific food item
- Strength of the statistical association
- Dose-response relationships
- The food’s production, preparation and service
- The food’s distribution
Food testing can provide useful information and help to support a hypothesis. Finding bacteria with the same DNA fingerprint in an unopened package of food and in the stool samples of people in the outbreak can be convincing evidence of a source of illness. However, relying on food testing can also lead to results that are confusing or unhelpful. This is the case for several reasons:
- Food items with a short shelf life, such as produce, are often no longer available by the time the outbreak is known, so they cannot be tested.
- Even if the actual suspected food is available, the pathogen may be difficult to detect. This is because the pathogen may have decreased in number since the outbreak or other organisms may have overgrown the pathogen as the food started to spoil.
- The pathogen may have been in only one portion of the food. A sample taken from a portion that was not contaminated will have a negative test result. So, a negative result does not rule out this food as a source of illness or the cause of the outbreak.
- Leftover foods or foods in open containers may have been contaminated after the outbreak or from contact with the food that actually caused the outbreak.
- Some pathogens cannot be detected in food because there is no established test that can detect the pathogen in the suspect food.
Sometimes in testing hypotheses, investigators find no statistical association between the illnesses and any particular food. This is not unusual, even when all the clues clearly point to foodborne transmission. In fact, investigators identify a specific food as the source of illness in about half of the foodborne outbreaks reported to CDC.
Not finding a link between a specific food and illness can happen for several reasons. Public health officials may have learned of the outbreak so long after it occurred that they could not do a full investigation. There may have been competing priorities or not enough staff and other resources to do a full investigation. An initial investigation may not have led to a specific food hypothesis, so no analytic study was done or the initial hypothesis could have been wrong. An analytic study may have been done, but it did not find a specific food exposure because the number of illnesses to analyze was small, because multiple food items were contaminated, or because the food was a "stealth food." Stealth foods are those that people may eat but are unlikely to remember. Examples include garnishes, condiments on sandwiches, and ingredients that are part of a food item (e.g., the filling in a snack cracker). Food testing did not find any pathogen related to the outbreak, or food testing may not have been done at all.
When no statistical association is found, it does not mean that the illness or outbreak was not foodborne. It means only that the source could not be determined. If the outbreak has ended, the source of the outbreak is declared unknown. If people are still getting sick, investigators must keep gathering information and studying results to find the food that is causing the illnesses.