• Write a null and alternative hypothesis for a hypothesis test.
• Decide if a sample statistic provides enough evidence to reject the null hypothesis.

The federal guidelines for water safety state that a city is compliant if at least [latex]90\%[/latex] of water samples obtained from residences have lead in the water under [latex]15[/latex] parts per billion (ppb). A water sample is “contaminated” if it contains lead at [latex]15[/latex] ppb or above. In other words, under [latex]10\%[/latex] of residences would need to be contaminated in order for the city to be compliant.

The residents of Flint suspected that more than [latex]10\%[/latex] of the homes in the city had contaminated water, so they took their own sample of residences as part of the Flint Water Study (FWS) with support from scientists at Virginia Tech.

How likely would it have been for the FWS to obtain a sample with a proportion of contaminated residences as high as [latex]20\% = 0.20[/latex] if the city was actually compliant? We can use the sampling distribution of the sample proportion and the normal distribution to determine this probability.

Since the normal distribution is continuous, it does not make sense to consider the probability of obtaining a sample with exactly [latex]20\%[/latex] of the residences contaminated (which would be [latex]0[/latex]). So, since we’re interested in the fact that the sample proportion was that high, we consider how likely we are to get a proportion that high or higher. In other words, how likely are we to have a sample proportion fall in that high range of [latex]20\%[/latex] or more if the true population proportion is only [latex]10\%[/latex]?

In this case, if the actual proportion of contaminated residences was only [latex]10\%[/latex], the probability of obtaining a sample with [latex]20\%[/latex] or more of residences yielding contaminated samples would be:

[latex]P(\hat{p}\geq 0.2) = 0.000002[/latex]