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Hypothesis testing in statistics is all about making decisions based on data. It helps us determine whether the evidence from our sample supports a specific claim, or hypothesis, about a population. For this experiment, we seek to determine if a new serum affects leukemia survival times in mice.

The concept involves two hypotheses:

• The Null Hypothesis ( H_{0} ): It suggests that there is no effect from the treatment. In this case, it states that the mean survival times for treated and untreated mice are equal.
• The Alternative Hypothesis ( H_{1} ): It suggests there is a measurable effect from the treatment. Here, it claims the mean survival times differ between the two groups, indicating effectiveness.

To conclude whether the serum is effective, we must evaluate these hypotheses with statistical evidence.

Pooled variance is a technique used when comparing two groups assumed to have equal variances. It's an important step when conducting a t-test, as it combines sample variances from each group into a single measure.

Here's how it works: You take the weighted average of the variances from the two groups to get the pooled variance:\[S_{p}^{2}= \frac{\left( (n_{1}-1)S_{1}^{2}+ (n_{2}-1)S_{2}^{2}\right)}{n_{1}+n_{2}-2}\]- The formula uses the variance from each group, S_{1}^{2} and S_{2}^{2}, applied to the number of observations- It helps in minimizing bias and gives a standardized measure to conduct further testsCalculated pooled variance will be used in our test statistic formula to determine if differences in the samples support rejecting the null hypothesis.

The critical t-value is an essential part of hypothesis testing when using the t-test. It acts as a threshold, determining whether the observed data significantly differ from what was expected under the null hypothesis. This value depends on your chosen level of significance (usually 0.05) and the degrees of freedom.

To find the critical t-value, you will:

• Consider the significance level, often set at 0.05 for a 95% confidence level.
• Calculate the degrees of freedom by adding the sample sizes of the two groups and subtracting 2, (n_{1}+n_{2}-2) .
• Look up this value in a t-distribution table to find the critical value that separates results by chance from those that are statistically significant.

If the test statistic is larger than this critical value (in absolute terms), the null hypothesis can be rejected.

The null hypothesis ( H_{0} ) is a foundational component of hypothesis tests. It begins from the standpoint that no effect or difference exists. By assuming H_{0} is true, scientists ensure any observed effect must be significantly proven before claiming a discovery. In this exercise, the null hypothesis is that the serum has no effect on the survival rates of mice, or X_{1} = X_{2} .

Working with the null hypothesis allows for:

• A structured test to measure deviation from the expected results when near-zero effect is assumed.
• Utilizing statistical tools to challenge the baseline hypothesis and potentially accept an alternative view.

The entire process hinges upon this null assumption, with calculations aimed at rejecting it in favor of the alternative hypothesis only if enough evidence is found.

The alternative hypothesis ( H_{1} ) is what researchers are ultimately pondering or what they suspect might be true if the null hypothesis is rejected. For our tests, it claims a difference in the mean survival times of treated and untreated mice exists, suggesting the serum's efficacy.

Here’s what you should know about the alternative hypothesis:

• It represents a change, effect, or difference between groups.
• In this case, it bets on the new serum having a measurable impact.
• We aim to gather enough statistical evidence to reject H_{0} and accept H_{1} .
  
  

Statistical significance in results would, thus, support the alternative hypothesis by indicating outcomes differ meaningly from the null's prediction.