The federal government recently granted funds for a special...

The federal government recently granted funds for a special program designed to reduce crime in high-crime areas. A study of the results of the program in eight high-crime areas of Miami, Florida, yielded the following results. Has there been a decrease in the number of crimes since the inauguration of the program? Use the .01 significance level. Estimate the $p$ -value.

Key Concepts

Hypothesis Testing

Hypothesis testing is a formal procedure used in statistics to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. This procedure involves setting up a null hypothesis (typically representing no effect or the status quo) and an alternative hypothesis (indicating the presence of an effect), then using sample data to test these hypotheses.

Significance Level

The significance level, often denoted by alpha (?), is the threshold used in hypothesis testing to determine whether to reject the null hypothesis. It represents the probability of committing a Type I error, which is rejecting the null hypothesis when it is actually true. In this context, a .01 significance level means that there is a 1% risk of wrongly rejecting the null hypothesis.

P-value

The p-value is the probability of obtaining test results at least as extreme as the observed results, assuming that the null hypothesis is true. It quantifies the evidence against the null hypothesis; a smaller p-value indicates stronger evidence in favor of the alternative hypothesis. In significance testing, if the p-value is less than the set significance level (e.g., .01), the null hypothesis is rejected.

One-tailed Test

A one-tailed test is a type of hypothesis test used when the alternative hypothesis specifies that the population parameter is either greater than or less than a certain value, but not both. In the context of determining if a program has decreased crime, a one-tailed test is appropriate since the interest is specifically in a decrease (i.e., testing whether the parameter is less than a given baseline).

Transcript

00:01 So we have some information, and we're going to let d stand for the difference between after minus before, and they're looking at the crime after they've implemented some programs.

00:13 And when i took the difference this way, the values that i had were, and let me quick just give you those, and you can double check me.

00:22 I have negative 12, 0, negative 1, 1, negative 9, 1, negative 5.

00:30 Let me arrow down, negative 4, and that's it.

00:35 So we have a sample size of 8 differences.

00:40 And we want to find, we'll be assuming that the mean difference is equal to 0, and alternately that the mean difference is actually negative, meaning that this number after is smaller than before so that we think that it's decreased.

00:59 And we have our data.

01:01 We know we have eight values.

01:04 And let me quick put in my, give you my mean and my standard deviation for those distributions, or for that distribution of eight numbers.

01:14 So we have that the x bar for those values is negative 3 .625.

01:22 Our standard deviation for those eight values is 4 .8385.

01:29 And we want to find our test statistic.

01:32 And before we do that, we're also using a 1 % significance level.

01:36 So let's find that critical t value right now.

01:39 And so that critical t value, we want to put all of that 0 .01 down in this tail, since we're doing a one -tail test, and this is where we would be rejecting our null and saying that we have evidence that it's actually smaller, that it has decreased the number of crimes with that federal program.

01:57 And let's find what that t value is.

01:59 That t value that will have seven degrees of freedom and it will have 1 % in the lower tail is going to be negative.

02:06 And let me get that lined up, seven degrees of freedom.

02:10 And 1%, we have 2 .998.

02:13 So this will be negative 2 .998...

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