Comparing Two Independent Population Means

Posted: August 26th, 2021

Statistical Inference

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Statistical Inference

Question 1: Hypothesis Testing Case

Comparing Two Independent Population Means

            When two means are different, there is a difference in the populations. In this comparison, two main assumptions are set first before proceeding with the analysis (Wasserman, 2004; Severini, 2015). These assumptions are:

1. The populations are independent simple random samples
2. The successes and failures in each of the independent samples are at least five.

A hypothesis test for a comparison of two populations is a typical statistics case. Such a hypothesis test helps ascertain whether the estimated means are different or exhibits population proportion difference (Rao, 2006). The difference between the two population means adapts a normal distribution (Goos & Meintrup, 2016). Typically, the null hypothesis is applied to state similarity in the two proportions. For example, if we have population X and population Y, for a null hypothesis, we expect p’X to be equivalent to p’Y where p’ is the population means. Hence, the null hypothesis will be stated as H0: p’X = p’Y.

When implementing the test, a pooled mean p’e is used in the assessment. The formula is used to calculate the pooled mean:

Thus, the difference is depicted through the following distribution;

. Based on the equation, the test statistic will be as follows;

(Utts & Heckard, 2007)

Real-World Example

When considering two types of COVID-19 vaccination, a test was made to assess whether there is a difference in the way the adult patients reacted to the drugs. A sample of 25 participants that were randomly selected from a total population of 250 adults was found to exhibit 30 minutes hives after receiving vaccination A. Another sample of 15 participants randomly selected from the same population of 250 adults still showed 30 minutes hives when subjected to vaccination Y. At 99% confidence interval (1% significance), the hypothesis test was implemented as follows;

If X and Y are the vaccination for X and Y, respectively, then p’X and p’Y are the anticipated populations means. The random variable, that is: p’X – p’Y demonstrates the difference in the means for the adult patients that did not exhibit any reaction after 30 minutes when subjected to the two vaccinations. The hypothesis for the test would;

Null Hypothesis, H0: p’X=p’Xp’Y-p’Y=0 (There is no difference in the population means)

The alternative hypothesis, HA: p’X≠p’Xp’Y-p’Y≠0 (There is a difference in the population means)

The distribution of the tests is demonstrated through an examination of two binomial population means with a normal distribution as;

Then, 1 – p’e = 0.92. Therefore,

, whereby p’X – p’Y follows an estimated normal distribution. The p-value is calculated through normal distribution, that p-value =0.1404

The approximate mean for group X:

The approximate mean for group X:

Graphically demonstrated for p-value;

Figure 1: p-value for the case (manually obtained)

Thus, p’X –p’Y = 0.1 – 0.06 = 0.04. As shown, half of the p-value is below -0.04 and a half above 0.04. Comparing the cases, α and p-value, α = 0.01. The decision would be that, since α = p-value, do not reject the null hypothesis, H0. Implying that at a 1% significance level, their evidence in the analysis is not sufficient to show that a difference exists in the two population means.

Question 2: Importance of Margin of Error

The margin of error refers to the plus or minus, in the case under question 1, it is indicated by confidence interval (99%) or range ±0.04. It acknowledges that there is variation in the sample results as well in the population condition (Bajpai, 2009). Specifically, it helps the analyst establish theextent of variation in the statistical results.

References

Bajpai, N. (2009). Business statistics. Delhi Upper Saddle River, N.J: Pearson.

Goos, P. & Meintrup, D. (2016). Statistics with JMP: hypothesis tests, ANOVA, and regression. Chichester, West Sussex: Wiley.

Rao, C. (2006). Linear statistical inference and its applications. New York: Wiley.

Severini, T. (2015). analytic methods in sports: using mathematics and statistics to understand data from baseball, football, basketball, and other sports. Boca Raton: CRC Press.

Utts, J. & Heckard, R. (2007). Mind on statistics. Belmont, CA: Duxbury, Thomson Brooks/Cole.

Wasserman, L. (2004). All of the statistics: a concise course in statistical inference. New York: Springer.