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Hypothesis testing is a method used in statistics to determine whether there is enough evidence in a sample to support a certain belief or hypothesis about the population. It's often used to test the effects of treatments, changes, or conditions.
There are usually two hypotheses involved in hypothesis testing: the null hypothesis (H_0) and the alternative hypothesis (H_1).

• The null hypothesis (H_0) represents a statement of no effect or no difference. For this exercise, it's that the median is equal to 68.
• The alternative hypothesis (H_1) represents what we want to prove. For instance, the median is different from 68.

When conducting a hypothesis test, we'll gather evidence from the data and determine if this evidence strongly contradicts H_0. If it does, we might reject H_0 in favor of H_1. If not, we do not reject H_0. This brings us to assessing the data using the sign test, detailed above.

The binomial distribution is a probability distribution that summarizes the likelihood that a value will take one of two independent states and is particularly useful in hypothesis testing.
In a sign test, we often refer to the binomial distribution because we're dealing with binomial trials – that is, trials that have only two possible outcomes (plus sign or minus sign).
To articulate it further:

• Let $$n$$ represent the total number of observations (in our example, 72).
• Let $$p$$ be the probability of success on each trial (this is usually 0.5 if the null hypothesis is true, indicating an equal chance of getting a plus sign or minus sign).

The critical values from the binomial distribution table, which depend on $$n$$and n$$p$$, help us ascertain whether our test statistic (the count of the lesser number of signs) falls into the rejection region or not.
This is why, in our example, the critical value at a 0.05 significance level for $$n = 72$$ and $$p = 0.5$$ was found to be approximately 26.

The significance level ($$\rna$$) in hypothesis testing is the threshold we set for deciding when to reject the null hypothesis. It is denoted by $$\rna$$ and usually set at values like 0.05, 0.01, or 0.10.
A significance level of 0.05, for example, implies that we would reject the null hypothesis if there's less than a 5% chance of observing our sample results (or more extreme) if the null hypothesis were actually true.
In this exercise, our chosen significance level is 0.05, meaning we're only willing to accept a 5% chance of making a Type I error (wrongly rejecting the null hypothesis).
Since our test statistic of 27 was not less than the critical value of 26, we do not reject $$H_0$$ . Thus, we conclude there isn't sufficient evidence at the 0.05 significance level to state that the median is different from 68.