Consider the following hypothesis test: H0 : μ ≥45 Ha : μ <45 A sample of 36 is used. I

step 1 For question 25, we're asked to consider the following hypotheses. So the no hypothesis is popul

Consider the following hypothesis test: H0 : μ ≥45 Ha...

Key Concepts

Significance Level (alpha)

The significance level, denoted by alpha (?), is the threshold for making a decision in hypothesis testing. It represents the maximum acceptable probability of committing a Type I error—rejecting the null hypothesis when it is actually true. Common levels include 0.05 and 0.01, and in a decision process, if the p-value is less than ?, the null hypothesis is rejected.

One-Tailed Test

A one-tailed test is used when the alternative hypothesis specifies a direction of the effect (e.g., less than or greater than a particular value). In this context, the test focuses on one end of the distribution of the test statistic, which makes it more powerful in detecting an effect in that specified direction compared to a two-tailed test.

p-value

The p-value is the probability, under the assumption that the null hypothesis is true, of observing a result at least as extreme as the test statistic calculated from the sample data. It quantifies the strength of the evidence against the null hypothesis, with smaller p-values indicating stronger evidence for the alternative hypothesis.

One-Sample t-Test

The one?sample t-test is used when the population standard deviation is unknown and must be estimated from the sample. It involves comparing the sample mean to a hypothesized population mean using the t-distribution, which accounts for the extra variability introduced by estimating the standard deviation from the sample.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves formulating a null hypothesis (typically representing a status quo or no effect) and an alternative hypothesis (representing a suspected effect or difference), and then using sample evidence to decide whether to reject the null hypothesis.

t-Distribution

The t-distribution is a probability distribution that is symmetric and bell-shaped, similar to the normal distribution, but with heavier tails. It is used in hypothesis testing when the sample size is small or the population standard deviation is unknown. The shape of the t-distribution depends on the degrees of freedom, generally determined by the sample size minus one.

Transcript

00:01 For question 25, we're asked to consider the following hypotheses.

00:04 So the no hypothesis is population mean is greater than or equal to 45.

00:10 And the alternative hypothesis is that new is less than 45.

00:16 And we're also told that we're dealing with samples of size 36 and that the confidence level for the question is alpha equals 0 .01.

00:29 Now, so then we have part a, part b, and part c.

00:32 In each of these parts, we're given a sample, and we're asked to calculate the p value and draw any conclusions with regard to our hypotheses.

00:40 So one thing to consider is that this is a one -sided test.

00:48 In particular, it's a lower -tail test.

00:55 And the reason that it's a lower -tail test is because the alternative hypothesis is mu is less than 45, so that's less than the no hypothesis.

01:08 And part a, we're given a sample average of 44 and a sample standard deviation of 5 .5.

01:18 And so the test statistic in this case is we don't have the population standard deviation.

01:26 So sample averages are distributed according to the t distribution.

01:30 So we're using the t test statistic.

01:34 And that is equal to the sample average minus population average divided by the sample standard deviation over the square root of the sample size.

01:44 And so for part a, this is 44 minus 45 over 5 .2.

01:53 Divided about square root of 36.

01:56 And this comes out to minus 1 .152.

02:01 And now we can use the t table to look up our p value.

02:05 But just a quick comment on this.

02:08 So suppose our symmetrical t distribution looks like this.

02:15 And normally when we look up the t value in the t table, it's giving us an area that is, it's giving us an upper -tailed area.

02:24 So it's all the area in the distribution above our t score.

02:28 But here we have a minus t score.

02:33 Let's say this is plus 1 .152.

02:37 So we're actually interested in the lower tailed area.

02:40 But due to symmetry, we can say that the area in this tail is identical to the area in this tail.

02:47 So we can treat it as if our t score is positive 1 .152 and then just take the upper -tailed area that's given in the table directly from the table.

03:02 I'll just delete this before moving on.

03:12 Okay, and then one more point to make is degrees of freedom is equal to n minus 1, which gives us 35...

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