Consider the following hypothesis test: H0: μ=15 Ha: μ≠15 A sample of 50 provided a

Consider the following hypothesis test: H0: μ=15 ...

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. It involves comparing a null hypothesis against an alternative hypothesis by analyzing test statistics and corresponding probabilities.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H?) represents a statement of no effect or no difference and is presumed true unless the data provide strong evidence against it. The alternative hypothesis (H?) represents what the researcher aims to support, typically stating that there is an effect or a difference.

Z-test

A Z-test is used for hypothesis testing when the population standard deviation is known and the sample size is typically large enough for the Central Limit Theorem to apply. It involves standardizing the sample mean to evaluate how many standard errors it lies from the hypothesized mean.

Test Statistic

The test statistic quantifies the difference between the observed sample statistic and the parameter stated in the null hypothesis, scaled by the standard error of the statistic. It provides a basis for determining how extreme the observed result is under the null hypothesis.

P-value

The p-value is the probability, assuming the null hypothesis is true, of obtaining a result at least as extreme as the one observed. It is used to assess the strength of the evidence against the null hypothesis: a smaller p-value indicates stronger evidence to reject the null.

Significance Level (?)

The significance level, denoted by ?, is a pre-specified threshold for determining whether the p-value is small enough to reject the null hypothesis. It represents the probability of making a Type I error, which is rejecting a true null hypothesis.

Critical Value and Rejection Rule

The critical value is the cutoff point on the test statistic distribution that defines the rejection region. If the test statistic falls beyond the critical value(s) in the rejection region, the null hypothesis is rejected. The rejection rule provides an alternative decision-making procedure to using the p-value for hypothesis testing.

Transcript

00:01 Okay, so we are given the following hypothesis test and the first thing we're asked to do is to calculate the test statistic.

00:07 So the formula for the test statistic is here, so we just take our sample mean, we subtract the mean from the null hypothesis and then we divide by the standard deviation of the sample mean.

00:21 So in this case, we're given all the values.

00:25 So we know from our non -hypothesis that mu -nought is 15.

00:30 Our observed mean is 14 .15.

00:33 Sigma, which is the standard deviation for the original population, is 3.

00:39 And n, which is a population size or the number of data points, is 50.

00:44 So if you just plug these into the formula, you get that the test statistic is equal to minus 2 .0, if you want to round it.

00:53 And then the next part of the question asks us to find a p value for this test statistic.

00:58 So what this is asking us is asking under the nile hypothesis, how likely is it that we observe a test statistic that's a value minus 2 or possibly more extreme than that? so you have to be a bit careful here because it's a two -tailed test.

01:16 So when we say a value is at least as extreme as extreme as minus 2, it could be less than minus 2 or it could also be bigger than plus 2.

01:25 So we need to calculate the probability that z is less than minus 2 or more than plus 2.

01:35 But because z is under the null hypothesis, it's a normally distributed random variable with mean zero and standard deviation 1.

01:47 We can just do this from the normal tables.

01:49 So like i just said, the probability that z is less than minus 2 or more than plus 2, is going to be equal to two times the probability that this normal random variable is less than minus 2 because the normal distribution is symmetrics.

02:04 The probability that it's at least plus 2 is the same as a probability that it's at least minus 2.

02:10 So you can read from your normal tables that the probability of this happening is 0 .0 .0275.

02:19 So if you time this by 2, you get 0 .0455 as your p value...

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