The 2007 global economic recession left devastating financial problems with many companies. After failing to recover from the effects of the recession, many firms turned to cost cutting as the main business strategy. In this regard, a tour company called company A uses cars made in the United States to transport clients. The firm wishes to cut costs by using cars with low fuel consumption.
To achieve this, the firm’s management launched a survey to determine where they could purchase cars with low fuel consumption. After consulting with players in the motor industry, it was revealed that cars made in Japan consume less fuel than cars made in the United States. To find out if this allegation was true, company A decided to collect data regarding fuel consumption for cars made in the US and Japan.
Research question: is fuel consumption for cars made in the US different from fuel consumption for cars made in Japan?
Since the researcher does not know the category of cars that consumes more fuel, a two-sided test is appropriate (Hazewinkel, 2001). The null hypothesis will be: Cars made in the US consume the same amount of fuel as cars made in Japan. This is tested against alternative hypothesis which says that the two categories of cars do not consume the same amount of fuel.
- H1: µ1 = µ2, against
- H2: µ1 ≠µ2.
The test involves two samples which are independent from each other (Hazewinkel, 2001). In each sample, there are two variables to be measured. The independent variable will be the number of gallons and the dependent variable will be the number of miles covered. This means that the number of miles covered depends on the number of gallons available. The cars that cover more miles with one gallon use less fuel.
Since the study involves comparing means of two populations, t-test or z-test can be used (Kaye & Freedman, 2011). But z-test can only be used if population means and standard deviations are known (Kaye & Freedman, 2011). In this case, both the means and standard deviations for the two populations are unknown.
This means that z-test cannot be used. Therefore, t-test is used. To use t-test, the variables are assumed to be normally distributed and the samples independent.
In the analysis, t-value will be determined from the data of collected samples. T-critical will be obtained from the t-table using (n1+n2-2) degree of freedom and a significance level of 0.05 (Kaye & Freedman, 2011). The null hypothesis is rejected if the absolute value of t is greater than t-critical (Kaye & Freedman, 2011).
From the two sets of data collected, the following results were obtained
For the first sample (Cars made in the US), sample size n1 = 248. Mean fuel consumption ẋ1 = 20.05 miles per gallon. Standard deviation s1 = 6.311. Standard error = s1/sqrt (n1) = 6.311/sqrt248 = 0.4007.
For the second sample (cars made in Japan), the sample size n2 = 78. Mean fuel consumption ẋ2 = 30.381. Standard deviation s2 = 6.008. Standard error = s2/sqrtn2 = 6.008/sqrt78 = 0.680.
T =(ẋ1 – ẋ2)/sp(sqrt [(1/n1) + (1/n2)]), where sp is the pooled standard deviation. sp² = [(n1-1)s1² + (n2-1)s2²] / (n1 + n2 – 2) =[(249-1)6.311²+(79-1)6.008²]/(248+78-2) =[(248)39.8287+(78)36.0961]/324=39.1760. Therefore, sp = 6.2591. This means that t = (30.381 – 20.05) / 6.2591(sqrt[(1/248) + (1/78)]) = -12.7143. The degree of freedom, df = n1+n2-2 = 248+78-2 = 324.
At a significance level of 0.05, t-critical = 1.6496 (from t-table). The null hypothesis is rejected since the absolute value of t = 12.7143 is greater than t-critical = 1.6496.
Therefore, the alternative is accepted. Fuel consumption for cars made in the US is different from fuel consumption for cars made in Japan. The observed difference in means was not due to chance or random error. Therefore, company A can proceed and acquire cars from Japan.
References
Hazewinkel, M. (2001). Student’s test. Journal of Mathematicsand Statistics. 5(2): 5-9.
Kaye, D. H. & Freedman D. A. (2011). Reference Guide on Statistics. Washington D.C: West National Academies.