ONE-SAMPLE HYPOTHESIS TESTING FOR POPULATION MEAN IN COMMUNICATION SCIENCE (2)

In a previous post, we explained how statistical hypothesis testing was conducted. In principle, there are four major steps, namely 1) formulate H₀ and H₁, 2) determine the level of significance, 3) determine the appropriate test statistic and the H₀ rejection area, 4) draw the conclusion. The beginning of this article contains important notes related to the steps.

About the formulation of null hypotheses and alternative hypotheses
In principle, the permitted relation signs in H₁ are one of the signs <, >, and ≠. Meanwhile, the possible signs of relations in H₀ are ≥, ≤, and =. The relation shown in H₁ is the negation of the relation shown in H₀. Thus, the permitted pairs are as follows.
H₀: θ ≥ θ₀
H₁: θ < θ₀
or
H₀: θ ≤ θ₀
H₁: θ > θ₀
or
H₀: θ = θ₀
H₁: θ ≠ θ₀

Here, θ denotes the population parameter being tested, and θ₀ is the hypothesized value of θ; θ₀ is a real number.

If the relation sign in H₁ is < or >, then the test is called a one-tailed test. The < sign indicates a left-tailed test, while the > sign indicates a right-tailed test. If the ≠ sign is used in H₁, then the test is called a two-tailed test. Determining the type of test is crucial because it affects the determination of the rejection/critical area and the conclusions (Steps 3 and 4).

It should be added here that some literature states that the sign of relation used in H₀ is “=” for all cases. Thus, according to the literature that “adheres” to this version, the permitted pairs of hypotheses are as follows.
H₀: θ = θ₀
H₁: θ < θ₀
or
H₀: θ = θ₀
H₁: θ > θ₀
or
H₀: θ = θ₀
H₁: θ ≠ θ₀

Both the previous and this version can be used, although the previous version is more commonly found in social science research.

About determining the level of significance
In real research, the level of significance is determined by the researcher, but a small value is usually selected because the level of significance is the probability of falsely rejecting H₀ while the statement in H₀ is true. In statistical hypothesis testing, there are two types of errors. Type I error occurs if we reject H₀ while the statement in H₀ is true. Type II error occurs if we do not reject H₀ while the statement in H₀ is false. So, we can say the significance level is the probability that a type I error occurs.

In his book entitled Communication Research Statistics (2006), J. C. Reinard states that in communication science research, a significance level of 0.05 is commonly used. However, in problems regarding hypothesis testing, this value is usually known in the problem.

About determining the appropriate test statistic and the rejection area of H₀
In statistics, there are many test statistics, depending on the parameters being tested. If the mean is being tested, there are three possible test statistics, namely:
with degrees of freedom (t statistic),
, and
.
However, in practice, the t statistic is often used because in the real world, the population standard deviation (σ) is unknown. What can be calculated from the sample is the sample standard deviation (s). The use of the t statistic above can be done if the sample comes from a normally distributed population. Therefore, in practice, before using this statistic, we need to do a normality test. If the result of the normality test is not significant, then the t statistic can be used.

In this step, determining the rejection region is optional. If we apply the critical value in concluding, then determining the rejection area is mandatory. If we use the p-value in concluding, then determining the rejection region is no longer necessary. The p-value approach is often preferred because of its ease in drawing conclusions.

About drawing conclusions
There are two possible outcomes in statistical hypothesis testing, namely the rejection of H₀ or the acceptance of H₀. A hypothesis test is said to be significant if we succeed in rejecting H₀. A hypothesis test is said to be insignificant if we fail to reject H₀. Whether we reject or do not reject H₀ in our conclusion, we are always faced with the possibility of making an incorrect conclusion. The probability of making an error by rejecting H₀ while the statement in H₀ is true is equal to the level of significance (α). However, the probability of making an error by not rejecting H₀ while the statement in H₀ is false cannot be determined. It is the reason why if we fail to reject H₀, then the test is said to be not significant. It should be emphasized that when we fail to reject H₀, it does not mean that what H₀ states is true. When we fail to reject H₀, we can only conclude that the sampling result is “insufficient” or “incapable” of rejecting H₀. When we succeed in rejecting H₀, we are quite confident that the conclusion we made is correct; the probability of being incorrect is only as small as α. Note that we cannot be sure whether or not an error occurs when drawing the conclusion. What we may state is only the probability that a type I error occurs, which is α.

Technically, what criterion is used to reject H₀? The answer depends on the approach used. The criterion in the critical value approach: if the value of the test statistic obtained from the sample is in the rejection/critical area, then reject H₀. In other circumstances, do not reject H₀. The criterion in the p-value approach: if p < α, then reject H₀. In other circumstances, do not reject H₀.

Look at the solutions to Case 2 and Case 3 to better understand the steps above. For ease of solution, we will use the p-value approach in concluding.

Case 2
A public relations firm launched a campaign to increase awareness about a new environmental policy. They want to determine if the campaign was successful in raising public awareness. They hypothesize that the average level of awareness (measured on a scale of 1 to 10, with 10 being highest) after the campaign is higher than the pre-campaign average of 5.7. A survey of 150 individuals was conducted after the campaign. The survey results data are recorded in the following .sav file: (click here). Is there sufficient evidence to conclude that the campaign was successful in raising public awareness? Use a 5% significance level. Assume that a normality test has been performed and the result is not significant.

STEP-BY-STEP SOLUTION

Step 1: Formulate the hypotheses
H₀: μ ≤ 5.7 [The campaign failed to raise public awareness.]
H₁: μ > 5.7 [The campaign succeeded in raising public awareness.]

Step 2: Determine the significance level
Here, the significance level has been determined, which is 0.05.

Step 3: Select a test statistic
Based on the information given, the normality test suggests nonsignificance. Therefore, we select the statistic with degrees of freedom . SPSS uses the term One-Sample T Test to refer to the test statistic chosen.

Step 4: Drawing conclusions
The t-value from the sample is calculated by using the formula . The following table shows the SPSS (Ver. 21) output.

From the table, we get the t value obtained from the sample (t_computed) of 1.388.

Step 5: Drawing conclusions
To apply the p-value approach in drawing conclusions, we look at the Sig. (2-tailed) value shown in the One-Sample Test table above, which is 0.167. It applies to the two-tailed test (2-tailed). The test we are dealing with is one-tailed, as seen from the sign of relation used in H₁, i.e., “>”. Therefore, in this case, to get the p-value, we must divide the Sig. (2-tailed) value by 2. Thus, . Because p > α (see Step 2), we cannot reject H₀. This test is not significant. Accordingly, the evidence is not sufficient to conclude that the average level of public awareness is more than 5.7. The campaign was not proven to be successful in increasing public awareness.

Additional question
In Case 2, H₀ was not rejected. What is the probability of making an error when concluding this? What is the probability of accepting H₀ while the statement in H₀ is false?

Answer
We do not know. If a hypothesis test is not significant, then the probability of making an error in concluding cannot be determined. Why? Because we do not know the actual value of the population parameter. The probability of such error can only be calculated if the value of the population parameter is assumed.

Note
In statistics, the probability of error made by not rejecting H₀ while the statement in H₀ is false, namely the probability of a type II error, is denoted by β. Another measure in hypothesis testing besides α and β is the power of the test, denoted by 1-β. The power of a test is the probability of rejecting H₀, assuming a fixed alternative parameter value is true. (The alternative parameter value is different from that in H₀.) In short, it can be said that power measures the sensitivity of the hypothesis test in detecting differences or deviations.

Case 3
A researcher is investigating the impact of using emojis in formal business communication. It’s generally accepted that the average perceived professionalism score for written business communication (on a scale of 1 to 10, with 10 being highly professional) is 6.5. The researcher wants to determine if including emojis in business emails significantly alters the perceived professionalism, either positively or negatively. A sample of 85 business professionals was asked to rate the professionalism of an email containing emojis. The sampling results are recorded in the following .sav file: (click here). Can it be concluded that the inclusion of emojis in business emails changes perceived professionalism? Use a significance level of 0.05. Assume that a normality test has been performed and the result of the test is not significant.

STEP-BY-STEP SOLUTION

Step 1: Formulate the hypotheses
H₀: μ = 6.5 [The inclusion of emojis in business emails does not change perceived professionalism.]
H₁: μ ≠ 6.5 [The inclusion of emojis in business emails changes perceived professionalism.]

Step 2: Determine the significance level
Here, the significance level has been determined, which is 0.05.

Step 3: Select a test statistic
Based on the information given, the normality test suggests nonsignificance. Therefore, we select the statistic with degrees of freedom .

Step 4: Drawing conclusions
The t-value from the sample is calculated by using the formula . The following table shows the SPSS (Ver. 21) output.

From the table, we get the t value obtained from the sample (t_computed) of 2.137.

Step 5: Drawing conclusions
To apply the p-value approach in concluding, we look at the Sig. (2-tailed) value shown in the One-Sample Test table above, which is 0.035. This value applies to a two-tailed test, appropriate for this test because in H₁ we use the relation sign ≠; the test being conducted is two-tailed. Thus, in this case, p = 0.035. Since p < α (see Step 2), we reject H₀. Accordingly, the average perceived professionalism score for written business communication is significantly different from 6.5. The inclusion of emojis in business emails significantly changes perceived professionalism.

Additional question
In Case 3, H₀ was rejected. What is the probability of incorrectly rejecting H₀ while it is true that μ = 6.5?
Answer
The probability of such an error occurring is 5%, which is the significance level.

MID-SEMESTER ASSESSMENT

1. Determine the population and quantitative variables to be tested using one-sample hypothesis testing for the population mean. For example, the intensity of social media use per day among college students, the duration of gadget use per day among elementary school students, the tuition fees in Bandung, and so on.
2. Determine the null hypothesis (H₀): μ = μ₀, μ ≥ μ₀, or μ ≤ μ₀ and the appropriate counter hypothesis (H₁): μ ≠ μ₀, μ < μ₀, or μ > μ₀
3. Determine the level of significance (α). It is usually set at α = 0.05 in communication science research.
4. Determine the appropriate test statistic and its critical region.
5. Calculate the value of the test statistic from the sample.
6. Give a decision on whether H₀ is rejected or not.
7. Provide a concluding sentence based on the decision given (Step 6 above).

Note:

Students are allowed to use the p-value approach to decide whether H₀ is rejected or not. If this approach is used, determining the critical region is no longer necessary.

The assessment’s answer sheet can be downloaded here: (click here)