Applying the Concepts

Applying the Concepts
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Background
Both descriptive and inferential statistics play a key role in aiding decision-making across different situations (Lehmann, 1997). Such statistical measures are also important in addressing quality concerns with products and services (Nickerson, 1997, Branch, 2014). The present article explored the role of such statistical measures in addressing quality concerns for a Minivan manufacturing company. As a responsible professional, my job is to ensure that the interests of the consumers are protected under all legible circumstances. Recently, the end-users of the minivans were complaining that the on-road mileage that they were actually getting differed from the mileage that was assured by the company before purchase of the minivan. The company claimed that the minivans would give a mean gas mileage of 28 miles per gallon. However, I inspected the manufacturing plant and randomly selected 41 new Minivans to test their on-road mileage. I tested the on-road mileage for each of the 41 new Minivans and observed that the mean mileage achieved by the new Minivans was 26.7 + 4.2 miles per gallon. The report is provided on different descriptive and inferential statistics. All statistical analysis for the present study was conducted with the in-silico.net/tools/statistics/ztest (professional version) software.
Results & Analysis
Part-One: Descriptive Statistics
Descriptive statistics results
Parameter Value
Mean 26.

700
SD 4.200
SEM 0.656
N 41     
90% CI 25.596 to 27.804
95% CI 25.374 to 28.026
99% CI 24.926 to 28.474
 
Table 1: Confidence-interval of the observed mean
As per table 1, the 95% confidence interval (CI) for the observed mileages is 25.37 to 28.026. Such findings suggest that the mean mileage claimed by the minivan company certainly falls within the 95% CI. The results signify that out of the 100% minivans tested, 95% minivans would provide a mileage between 25.37 and 28.026 miles per gallon.
Part Two: One-sample Two-tailed z-test
The tested mean mileage was compared to the claimed mileage by a one sample two-tail z-test. The test was conducted to estimate whether the observed mileage significantly differs from the claimed mileage. The null hypothesis (Ho) and the alternative hypothesis (Ha) for the research questions were (Nickerson, 1997, Branch, 2014):
Ho = there is no significant difference between the mean mileage that is actually achieved by the new minivans and the mean mileage that is claimed by the manufacturers of the minivan. Any observed difference between the mean mileage that is actually achieved and that claimed by the company could be attributed to the chance factors that are associated with random sampling. The null hypothesis would be accepted if the p-value for the statistical test of inference (one-sample two-tailed z-test) is greater than 0.05 (p>0.05).
Ha = there is a significant difference between the mean mileage that is actually achieved by the new minivans and the mean mileage that is claimed by the manufacturers of the minivan. Any observed difference between the mean mileage that is actually achieved and that claimed by the company cannot be attributed to the chance factors that are associated with random sampling. The alternative hypothesis would be accepted if the p-value for the statistical test of inference (one-sample two-tailed z-test) is lesser than 0.05 (p<0.05).
Result
z-value -1.9819
two-tailed p-value 0.0475
 
95% confidence intervals  
upper 29.2856
lower 26.7144

The two samples are significantly different.
Table 2: Outputs of one-sample two-tailed z-test
Table 2 reflects that the observed value of the test statistic (z) is -1.9819 while the critical z value at the 0.05 level of significance at df (degrees of freedom)=40 is -1.644. The observed test statistic certainly lies is the critical region. Hence, the p-value for the one-sample two-tailed z-test is 0.0475. Since the p-value for the one-sample two-tailed z-test is less than the chosen level of significance (0.05), the null hypothesis is rejected and the alternative hypothesis is accepted. This means that there is a significant difference between the mean mileage that is actually achieved by the new minivans and the mean mileage that is claimed by the manufacturers of the minivan. Hence, the company’s claim is rejected.
Part Three: One-sample one-tailed z-test
The tested mean mileage was compared to the claimed mileage by a one sample two-tail z-test. The test was conducted to estimate whether the observed mileage is significantly lower or higher than the claimed mileage. The null hypothesis (Ho) and the alternative hypothesis (Ha) for the research questions were(Nickerson, 1997, Branch, 2014):
Ho = the mean mileage that is actually achieved by the new minivans is not significantly lesser than the mean mileage that is claimed by the manufacturers of the minivan. Any observed reduction in the mean mileage that is actually achieved compared to that claimed by the company could be attributed to the chance factors that are associated with random sampling. The null hypothesis would be accepted if the p-value for the statistical test of inference (one sample z-test) is greater than 0.05 (p>0.05).
Ha = the mean mileage that is actually achieved by the new minivans is significantly lesser than the mean mileage that is claimed by the manufacturers of the minivans. Any observed reduction in the mean mileage that is actually achieved compared to that claimed by the company could not be attributed to the chance factors that are associated with random sampling. The alternative hypothesis would be accepted if the p-value for the statistical test of inference (one sample z-test) is lesser than 0.05 (p<0.05).
Result
z-value -1.9819
one-tailed p-value 0.0237
   
95% confidence internvals  
upper 29.0789
lower -Inf
The two samples are significantly different.
Table 3: Outputs of one-sample one-tailed z-test
Table 2 reflects that the observed value of the test statistic (z) is -1.9819 while the critical z value at the 0.05 level of significance at df (degrees of freedom)=40 is -1.644. The observed test statistic certainly lies is the critical region. Hence, the p-value for the one-sample two-tailed z-test is 0.0237. Since the p-value for the one-sample two-tailed z-test is less than the chosen level of significance (0.05), the null hypothesis is rejected and the alternative hypothesis is accepted. This means that the mean mileage that is actually achieved by the new minivans is significantly lesser than the mean mileage that is claimed by the manufacturers of the minivans. Hence, the company’s claim is once again rejected. On the other hand, the customers’ complaints should be accepted.
Conclusion The two methods for checking the means are equally important. Initially, two-tailed tests should be conducted to explore whether there is significant difference between two means. If there is significant difference between two means then a one-tailed test should be conducted. The one-tailed tests would help to portray whether one of the means is significantly higher or lower than the compared mean. The chances of type-1 error are higher with one-tailed tests. My analysis reflect that the company’s claim is not true and therefore; the company should take stringent measures for ensuring the quality of mileage of the newly manufactured minivans.
References
Branch, M (2014). “Malignant side effects of null hypothesis significance testing”. Theory & Psychology. 24 (2), 256–277
 Lehmann E (1997). “Testing Statistical Hypotheses: The Story of a Book”. Statistical Science. 12 (1), 48–52
Nickerson, S. (2000). “Null Hypothesis Significance Tests: A Review of an Old and Continuing Controversy”. Psychological Methods. 5 (2), 241–301.