A bulb manufacturer claims that brand A bulbs last longer...

A bulb manufacturer claims that brand A bulbs last longer than brand B bulbs. To test this claim, the life of both brands of bulbs, per hour, were recorded: $$\begin{array}{l|cccccc}\text { Brand A } & 810 & 850 & 760 & 690 & 820 & 830 \\\hline \text { Brand B } & 765 & 720 & 700 & & &\end{array}$$ Use the rank-sum test with $\alpha=0.05$ to test whether the claim is valid.

A bulb manufacturer claims that brand A bulbs last longer than brand B bulbs. To test this claim, the life of both brands of bulbs, per hour, were recorded:
$$\begin{array}{l|cccccc}\text { Brand A } & 810 & 850 & 760 & 690 & 820 & 830 \\\hline \text { Brand B } & 765 & 720 & 700 & & &\end{array}$$
Use the rank-sum test with $\alpha=0.05$ to test whether the claim is valid.

Key Concepts

Nonparametric Tests

Nonparametric tests are statistical procedures that do not assume a specific distribution for the data, making them suitable for small samples or non-normal populations. These tests largely use ranks or medians instead of means and variances, allowing for a more robust analysis when traditional parametric assumptions may be violated.

Rank-Sum Test (Mann-Whitney U Test)

The rank-sum test, also known as the Mann-Whitney U test, is a nonparametric method used to compare two independent groups by ranking the combined data and examining the sums of these ranks. It tests whether one group tends to have higher scores than the other, making it useful when data do not meet the assumptions of parametric tests like the t-test.

Hypothesis Testing

Hypothesis testing involves formulating a null hypothesis, which assumes no effect or difference, and an alternative hypothesis that represents the effect or difference of interest. The process uses a test statistic computed from the data, and decisions are made by comparing this statistic against critical values or by evaluating the p-value.

Significance Level and P-Value

The significance level (?) is the pre-determined threshold for rejecting the null hypothesis, reflecting the risk of making a Type I error. The p-value is the probability of observing test statistics as extreme as those recorded, assuming the null hypothesis is true. Together, they help determine whether the observed data provide sufficient evidence to support the alternative hypothesis.

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