Statistical Analysis: Difference In Battery Life Of Energizer And Ultra-cell Batteries

Descriptive Values for Both Batteries

1. Units: 18

Treatment Variable: Choice of batteries – Energizer and Ultra-cell

Response Variable:  Playing time in hours

Two independent groups have to be tested by independent t-test. Let and be the average battery life of Energizer and Ultra-cell batteries.

Summary statistics table is as follows,

Table 1: Descriptive Values for Both Batteries

Batteries	Mean	Std. deviation	n
Energizer	8.2789	0.2174	9
Ultra-cell	8.2433	0.1628	9

Independent t-test

1. -= the difference between average playing life time in hours of Energizer and Ultra-cell batteries.
2. Null Hypothesis: H0: -=0
3. Alternate Hypothesis: H1: -0 (two-tailed)
4. From summary statistics of the sample survey, = 8.2789 – 8.2433 = 0.0356
5. E () = 0.0905 (from t?procedures tool on Canvas) and the test statistic was calculated using the formula aswith 9 degrees of freedom.
6. P-value = (from t?procedures tool on Canvas)
7. P-value interpretation: The p-value was greater than 0.05, and there was not enough support of H1 against the null hypothesis H0. The observed difference in average playing hours for the two type of batteries (D = 0.0356 hours) was not statistically significant at 5% level of significance. Hence, due to lack of sufficient evidence the null hypothesis could not be rejected. Therefore, there was no statistically significant difference between averages of playing time due to both the batteries (Hinton, 2014).

1. Approximate confidence interval at 5% level of significance for -was calculated as, where t-multiplier was obtained from t?procedures tool on Canvas.

1. Confidence Interval elucidation: With 95% probability or confidence it can be stated that the average battery hours of Energizer batteries would be approximately anywhere between 0.17 hours less than and 0.24 hours more than average battery hours of Ultra-cell batteries. The right hand limit of the confidence interval of 0.24 hours or approximately 15 minutes would be practically significant result.

1. Conclusion: From the sample data of 9 batteries of both the brands, not enough evidence was found to establish any significant difference in battery life for playing electronic game. Though, average battery life of Energizer batteries was greater than that of ultra-cell batteries, the difference was not statistically significant to opt for any particular brand.

1. The true value of the parameter was the hypothesized difference in average battery life of Energizer and Ultra-cell batteries (which was zero). The value of the parameter was well within the confidence interval (at 5% level of significance), indicating that the conclusion from p-value was true. The null hypothesis could not be rejected.

1. The sampling situation for the particular scenario was: One sample with multiple inclusive response categories.
2. The sampling for scrutinizing the difference between the estimated proportions of responses supporting legalizing cannabis- based products and feeling that the law should stay unchanged was tested by t-test for difference between two proportions.

1. Let and denote the proportions of responses supporting and refuting the legislation. Hence, -denotes the difference of the two above mentioned proportions.
2. Null hypothesis: H0: -= 0
3. Alternate hypothesis: H1: -(two tailed)
4. Estimated difference: -=
5. The test statistic formula used

For estimated difference = 0.61 and hypothesized difference = 0, the value of the standard error was calculated using t?procedures tool on Canvas as S.E (-) = 0.0333 at 5% level of significance. So the test statistic was calculated as with degrees of freedom =

1. P-value = (from t?procedures tool on Canvas)
2. Interpretation of p-value:

The p-value was greater than 0.05 at 5% level of significance and there was enough evidence in favor of the alternate hypothesis against the null hypothesis. Hence, at 5% level of significance, evidences from the difference in proportions of adult New Zealanders in support and against the legislation of cannabis?based products usage for medicinal purposes were sufficient to reject the null hypothesis.

1. The confidence interval was calculated as where t-estimate = 1.96 was obtained from t?procedures tool on Canvas.
2. Interpretation of Confidence Interval:

The estimated value of difference between the proportions of views in support and in against of adult New Zealanders, with 95% confidence, should somewhere between 0.5447 and 0.6752. The limits also signified that views in support were higher than views in against by 54.47% to 67.52%, indicating the practical significance of the confidence interval limits.

1. Conclusion:

The claim in the null hypothesis was rejected, which signified that proportion of responses of adult New Zealanders in favor of the legislation was significantly different (greater) than that of the responses of adult New Zealanders against the legislation.

1. i) The hypothesized value of difference in productivity score was zero.

(ii) The estimated difference in productivity score (Case 1) was greater than (right hand side) the hypothesized value at 5% level of significance. The standard deviation of the sampling distribution (standard error) was 0.682, indicating that the estimated values of the difference in productivity scores were not close to the hypothetical value. Hence, the hypothesized value was observed to be outside the confidence interval of the estimated difference, which was constructed using the standard error.

1. At 5% level of significance, all the cases excluding Case 3 demonstrated that the sample mean difference was statistically significant.
2. (i) In Case 1 the sample mean difference was practically significant.

(ii) In Case 4 and Case 5 the sample mean differences were not practically significant.

1. In Case 2 and Case 3 sufficient evidence of practically significant mean difference was not available. Hence, the nothing could be concluded from the confidence intervals of Case 2 and Case 3.
2. The observed mean difference in Case 6 was statistically significant at 5% level of significance. With 95% confidence, it was inferred that the estimated mean difference of the two payout systems would be somewhere between 0.96 hours to 5.58 hours. If the actual difference of the means of the two pay-out systems would be as low as 0.96 hours and high as 5.58 hours. In both cases, the results would be practically insignificant considering the management’s decision. The model in Case 6 was statistically significant, but practically insignificant. Statistically, Bonus pay-out system would be preferred. But, considering the management’s consideration about the difference in productivity score the company would like to stick to its previous model for pay-out.

1. (i) Units were 12 short stories from mystery, ironic, and literary fields.

(ii) Treatment was inclusion or exclusion of spoiler paragraph.

(iii) Response variable was the enjoyment rating of the readers.

1. (i) The graphs of the two treatments and their difference were constructed using iNZight and have been represented.

(ii) From side-by-side box plots, it was evident that median of enjoyment scores of the stories was greater in case of spoiler paragraph in front of the story. The median enjoyment score for stories with spoiler was nearly 7.0, whereas the median score of enjoyment for the stories without spoiler paragraph was near the 6.0 mark. With spoiler paragraph the distribution of enjoyment scores was highly left skewed compared to that of the scores without spoiler paragraph.

The box plot of difference in the enjoyment scores were plotted and has been represented in Figure 4. The median of difference in enjoyment scores was around 0.5, and the distribution was observed to follow Gaussian distribution.

1. The difference between average enjoyment scores for the two treatments was verified by t-test as follows.

The descriptive values for both treatments have been provided in Table 2.

Table 2: Descriptive Values for Enjoyment Scores

Treatment	Mean	N	Std. Deviation	Std. Error   Mean
With spoiler	6.217	12	1.2202	.3522
No spoiler	5.725	12	1.2563	.3627

Parameter: Let and are the average enjoyment scores for stories with and without spoiler paragraph. Hence, -denotes the difference in average enjoyment scores.

1. Null hypothesis: H0: -= 0
2. Alternate hypothesis: H1: -(two-tailed)
3. Estimate: The difference in average enjoyment scores from the sample data was
4. Test statistic: where standard error SE = 0.1003 (from SPSS) with 11 degrees of freedom.
5. P-value = (from SPSS) (Cronk, 2017)
6. Interpretation of P-value:

Independent t-test

There was very strong evidence against the null hypothesis, and average enjoyment score differences for 12 stories was found to be significantly different for two treatments. Average enjoyment score in case of spoiler paragraph added to the story was significantly (statistically) different (greater) than that of the stories without any spoiler paragraphs at 5% level of significance.

1. Confidence Interval:

Approximate confidence interval at 5% level of significance for – was calculated as, (from SPSS output).

1. Confidence Interval elucidation: With 95% probability or confidence it can be stated that the average enjoyment score of 12 stories with spoiler would be approximately anywhere between 0.27 less than and 0.71 more than average enjoyment score of 12 stories without spoiler. Both hand limits of the confidence interval would be practically significant result.

1. Conclusion: From the sample data of 12 stories for both the treatments, very strong evidence was found to establish significant difference in enjoyment score for reading stories. Average enjoyment score of stories with spoiler was greater than that of stories without spoiler. Hence, readers were enjoying reading stories with spoiler paragraph at front.

1. The dependent variable was the difference of enjoyment scores from two treatments. The difference scores were continuous in nature.

There were two categorical values of the treatment attribute.

The distribution of difference of enjoyment scores seemed to be normally distributed from Figure 4. But, using Shapiro-Wilk test in SPSS it was established that the differences of enjoyment scores were not normally distributed (SW = 0.955, p = 0.712). Therefore, the fourth condition for dependent t-test was not satisfied (Kim, 2015).

1. (i) Units were 1000 cyclists who completed the 180 kilometer ride event in the years 2010 to 2017.

(ii) Treatment was four age groups in the study.

(iii) Response variable was the time to complete K2.

1. (i) The graphs of the four treatments were constructed using iNZight and have been represented.

(ii) Time taken by the cyclists to complete the event of K2 Road Cycle Classic was noted to increase with the age of the cyclist. The median hours to complete the event increased for older age groups. From the spread of the box plots and distribution pattern of the histograms is was evident that due to increase in age, the variation in completion time also increased. Outliers were also noted for each age group.

1. SPSS output for F-test has been provided in Table 3.

Table 3: ANOVA Output from SPSS

Time in hours
	Sum of Squares	df	Mean Square	F	Sig.
Between Groups	64.327	3	21.442	28.104	.000
Within Groups	759.914	996	.763
Total	824.241	999

Assumptions: Event completion time of four age groups was independent of each other, completion time for the cyclists of all the four age groups was normally distributed, and the variances of the four age groups were significantly equal.

1. Smallest standard deviation was found for the 18-34 age group (M = 6.62, SD = 0.762) and the highest standard deviation was found for the 55+ age group (M = 7.32, SD = 0.913). The ratio was calculated as.
2. The completion time of the cyclists of four age groups was independent of each other. From Box plots of Figure 6, the symmetric nature of the distribution of completion times for the age groups were evident. From Shapiro Wilk test the distribution of completion times for the age groups were found to be normal (Appendix – Table 7). The assumption of homogeneity or equality of variances for event completion time of the four age groups was found to be true (L = 2.498, p = 0.058) (Appendix – Table 8) (Ross, & Willson, 2017).
3. (i) Null hypothesis: Variances in average event completion time for the four age groups were equal

H0:

(ii) Alternate hypothesis: Variances in average event completion time for the four age groups were significantly different.

(iii) Difference in average time taken by the cyclists for completion of the event were statistically significant (F = 28.104, p < 0.05) for the four age groups at 5% level of significance. The youngest cyclists were found to be more agile in completing the event compared to the older age groups, and the trend was also practically significant.

1. (i) From Tukey HSD post hoc analysis difference in average completion time between the age groups of 18-34 and 35-44 was not statistically significant (MD = – 0.336, p = 0.982) at 5% level of significance. The 95% confidence interval was found to be [-0.2665, 0.1992]. With 95% confidence, it was possible to claim that the average time (in hours) taken by the cyclists of age group 18-34 (years) was 0.27 hours less than, and 0.20 hours greater than that of the cyclists of 35-44 (years) age group (Murphy, Myors, & Wolach, 2014).

(ii) At 5% level statistical significance in pair wise difference of average completion time to complete K2 was noted for the age groups of 18-34 and 45-44, 18-34 and 55+, 35-44 and 45-54, 35-44 and 55+, and 45-54 and 55+.

(iii) At 5% level of significance the slowest of the four age groups were the cyclists of age group of 55+ years.

1. The cyclists took minimum of 5.45 hours and maximum of 11.29 hours to complete the K2 event. Age was a significant factor in completion time of the participants. Cyclists, aged between 18 years to 44 years were found to complete the event with almost identical average completion time. Cyclists, aged above 55 years were significantly slow compared to participants of other age groups. Though, in every age group some cyclists were found to complete the event with significant difference in completion time (outliers) (Mertler, & Reinhart, 2016).

1. (i) Scenario 1: Sex and Free

Scenario 2: First and Spent

Scenario 3: Age and Purchase

Scenario 4: Quantity_Stone and Quantity_Other

Scenario 5: Shop and Spent

(ii)

Table 4: Variable Type Details

Variable	Type
Sex	Categorical
Age	Numerical
First	Categorical
Spent	Numerical
Quantity_Stone	Numerical
Quantity_Other	Numerical
Shop	Categorical
Free	Categorical
Purchase	Categorical

The tools that could be used for the scenarios have been provided as below.

Table 5: Tools for Different Scenarios

Scenario	Tool(s)
Scenario 1	Table of counts     Side-by-side bar charts of proportions     Side-by-side stacked bar charts
Scenario 2	Side-by-side plots on the same scale
Scenario 3	Side-by-side plots on the same scale
Scenario 4	Scatter plot      Tile density plot
Scenario 5	Side-by-side plots on the same scale

c) The analyses which can be used for investigating the scenarios have been provided in the following table.

Table 6: Scenario with Test Matching

Scenario	Test
Scenario 1	E
Scenario 2	D
Scenario 3	F
Scenario 4	C
Scenario 5	F

References

Cronk, B. C. (2017). How to use SPSS®: A step-by-step guide to analysis and interpretation. Routledge.

Hinton, P. R. (2014). Statistics explained. Routledge.

Kim, T. K. (2015). T test as a parametric statistic. Korean journal of anesthesiology, 68(6), 540-546.

Mertler, C. A., & Reinhart, R. V. (2016). Advanced and multivariate statistical methods: Practical application and interpretation. Routledge.

Murphy, K. R., Myors, B., & Wolach, A. (2014). Statistical power analysis: A simple and general model for traditional and modern hypothesis tests. Routledge.

Ross, A., & Willson, V. L. (2017). One-Way Anova. In Basic and Advanced Statistical Tests (pp. 21-24). SensePublishers, Rotterdam.