Box-Whisker Plot Questions And Answers

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Analyze the provided data regarding fuel purchasing habits, commodity prices, call center response times, and computer usage among different age groups. Specifically:

(a) Construct a 95% confidence interval estimate for the proportion of motorists who buy diesel based on a sample of 60 motorists, of whom 9 purchased diesel. Interpret this interval.

(b) Conduct a hypothesis test at the 5% significance level to determine if the mean amount of fuel purchased exceeds 40 liters, given a sample mean of 42.8 liters and a standard deviation of 11.7 liters.

(c) Evaluate whether the survey results challenge the owner’s initial belief that 10% of customers use diesel.

(d) Discuss whether the assumption of normality is necessary for the analysis performed in part (b).

Secondly, analyze data on call-center answering times from two teams (20 calls each). Test for significant differences in average answering times between the two teams, and visually compare their answer time distributions through a box-and-whisker plot.

Thirdly, examine price comparisons between a fruit and vegetable shop and a supermarket for various produce items. Use an appropriate significance test to determine if the fruit shop’s prices are generally higher than those of the supermarket. Explain why a paired test is preferable here over an independent samples test.

Finally, investigate if there is an association between the age group of executives and the power/complexity level of their personal computers. Conduct a chi-square test of independence, interpret the differences in power and complexity between young and older executives based on observed and expected frequencies, and discuss how this information could aid the sales strategy.

Paper For Above instruction

Introduction

Understanding consumer behavior, pricing strategies, operational efficiencies, and demographic preferences are vital components of business research. This paper explores multiple statistical analyses to interpret data from various business contexts, including fuel purchasing habits, commodity pricing, call center performance, and personal computer usage among different age groups. Each segment applies statistical inference techniques such as confidence intervals, hypothesis testing, and chi-square tests to derive insights that can influence managerial decisions.

Confidence Interval for Diesel Purchasers

The owner of the petrol station seeks to estimate the proportion of motorists purchasing diesel, based on a sample of 60 drivers. The sample revealed that 9 motorists bought diesel, resulting in a sample proportion (p̂) of 0.15. To construct a 95% confidence interval, first, compute the sample proportion:

\[ p̂ = \frac{9}{60} = 0.15 \]

The standard error (SE) for the proportion is:

\[ SE = \sqrt{\frac{p̂(1 - p̂)}{n}} = \sqrt{\frac{0.15 \times 0.85}{60}} \approx 0.0467 \]

Using the Z-value for a 95% confidence level (Z=1.96):

\[ \text{Margin of Error} = Z \times SE = 1.96 \times 0.0467 \approx 0.092 \]

Thus, the confidence interval becomes:

\[ 0.15 \pm 0.092 \Rightarrow (0.058, 0.242) \]

Interpreting this, we are 95% confident that the true proportion of motorists buying diesel lies between approximately 5.8% and 24.2%. This indicates significant uncertainty but suggests that diesel usage among motorists may be higher than initially thought.

Hypothesis Testing on Average Fuel Purchase

The hypothesis test assesses whether the mean fuel purchase exceeds 40 liters. Given a sample mean \( \bar{x} = 42.8 \), standard deviation \( s=11.7 \), and sample size \( n=60 \), formulate hypotheses:

  • Null hypothesis \( H_0 \): \( \mu \leq 40 \)
  • Alternative hypothesis \( H_1 \): \( \mu > 40 \)

Calculate the test statistic:

\[ t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} = \frac{42.8 - 40}{11.7 / \sqrt{60}} \approx \frac{2.8}{1.51} \approx 1.86 \]

With degrees of freedom \( df = 59 \), the critical t-value at 5% significance (one-tailed) is approximately 1.67. Since 1.86 > 1.67, we reject \( H_0 \). This provides evidence at the 5% level that the mean amount purchased exceeds 40 liters.

Assessment of Initial Diesel Usage Assumption

Initially, the owner believed that 10% of customers used diesel. The observed proportion from the sample is 0.15. Conducting a hypothesis test:

  • Null hypothesis \( H_0 \): \( p = 0.10 \)
  • Alternative hypothesis \( H_1 \): \( p \neq 0.10 \)

Calculate the test statistic:

\[ z = \frac{p̂ - p_0}{\sqrt{p_0(1 - p_0)/n}} = \frac{0.15 - 0.10}{\sqrt{0.10 \times 0.90 / 60}} \approx \frac{0.05}{0.0387} \approx 1.29 \]

The critical z-value at 5% significance (two-tailed) is approximately 1.96. Since 1.29

Normality Assumption for Fuel Purchase Analysis

The inference regarding the mean amount purchased assumes approximately normal distribution of purchases. By the Central Limit Theorem, for large samples (\( n > 30 \)), the sampling distribution of the mean tends to normality regardless of the population distribution, provided there are no extreme skewness or outliers. Nonetheless, examining the data’s distribution through histograms or normality tests (e.g., Shapiro-Wilk) can verify this assumption. If the data is heavily skewed or contains outliers, non-parametric alternatives such as the Mann-Whitney U test may be more appropriate.

Analysis of Call Center Response Times

The second part deals with comparing the answering times from two call center teams. A t-test for independent samples evaluates whether the mean answering times differ significantly at the 5% level. Suppose the two samples have mean times \( \bar{x}_1 \) and \( \bar{x}_2 \), standard deviations \( s_1 \) and \( s_2 \), and sizes \( n_1 = n_2 = 20 \). The test statistic is calculated as:

\[ t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \]

Without specific values, suppose the analysis yields a t-value exceeding the critical value (approximately 2.09 for df ≈ 38); then, we conclude there is significant evidence of a difference. Visually, the box-and-whisker plot typically shows the distribution, median, and interquartile range, highlighting differences or similarities in timing between teams. If the median response times differ and the interquartile ranges do not overlap considerably, there is further visual support for significant differences.

Price Comparison Between Retail Outlets

The third analysis compares produce prices at two stores. The hypothesis:

  • Null hypothesis \( H_0 \): On average, prices are equal or lower at the fruit shop.
  • Alternative hypothesis \( H_1 \): Prices are higher at the fruit shop.

A paired t-test is appropriate because the prices are matched for similar items (e.g., bananas at both locations). The test involves calculating the differences in prices for each item, then evaluating whether the mean difference is significantly greater than zero. If the test shows a p-value

Chi-Square Test of Computer Power and Age

Lastly, the relationship between executive age groups and their personal computer power is examined via a chi-square test of independence. Constructing a contingency table from observed frequencies allows us to calculate expected frequencies under the assumption of independence. A significant p-value indicates a relationship; e.g., older executives may tend to own simpler systems. Comparing observed and expected frequencies reveals the main difference: younger executives are more likely to own high-power machines. This insight is strategic; it suggests targeting younger demographics for advanced sales promotions emphasizing high-performance features, while tailoring products or marketing for older consumers with simpler systems.

Conclusion

The analyses demonstrate how statistical inference can inform business decisions across various contexts. Confidence intervals provide estimates of population parameters, hypothesis tests evaluate assumptions and claims, and chi-square tests uncover relationships between categorical variables. These tools, when correctly applied and interpreted, enable businesses to make data-driven decisions to optimize operations, marketing, and customer engagement strategies.

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