Consider the following hypothesis test: H0: μ=18 Ha: μ≠18 A sample of 48 provided a

step 1 So in this problem, we're given the following information. That our null hypothesis includes a m

Consider the following hypothesis test: H0: μ=18 ...

Consider the following hypothesis test: $$ \begin{aligned} & H_0: \mu=18 \\ & H_{\mathrm{a}}: \mu \neq 18 \end{aligned} $$ A sample of 48 provided a sample mean $\bar{x}-17$ and a sample standard deviation $s=4.5$. a. Compute the value of the test statistic. b. Use the $t$ distribution table (Table 2 in Appendix B) to compute a range for the p-value. c. At $\alpha=.05$, what is your conclusion? d. What is the rejection rule using the critical value? What is your conclusion?

Consider the following hypothesis test:
$$
\begin{aligned}
& H_0: \mu=18 \\
& H_{\mathrm{a}}: \mu \neq 18
\end{aligned}
$$
A sample of 48 provided a sample mean $\bar{x}-17$ and a sample standard deviation $s=4.5$.
a. Compute the value of the test statistic.
b. Use the $t$ distribution table (Table 2 in Appendix B) to compute a range for the p-value.
c. At $\alpha=.05$, what is your conclusion?
d. What is the rejection rule using the critical value? What is your conclusion?

Key Concepts

Rejection Rule

The rejection rule defines the criteria based on which the null hypothesis is rejected in hypothesis testing. It can be based on critical values obtained from the t-distribution for a given significance level, or by comparing the p-value to the significance level. If the test statistic falls in the rejection region (or if the p-value is below the significance level), the null hypothesis is rejected.

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 formulating a null hypothesis, which represents a default position or a statement of no effect, and an alternative hypothesis, which represents the expected deviation or effect.

Null and Alternative Hypotheses

The null hypothesis (H0) is the assumption that there is no effect or difference, such as the population mean equaling a specified value, while the alternative hypothesis (Ha) suggests a change or difference (for example, the population mean is not equal to the specified value). These hypotheses set the basis for evaluating evidence from sample data.

Test Statistic

The test statistic is a standardized value calculated from sample data that is used to decide whether to reject the null hypothesis. It measures the degree of departure of the sample statistic from the null hypothesis value, relative to the variability in the data. In this context, the t-statistic is used when the population standard deviation is unknown.

t-Distribution

The t-distribution is a probability distribution used in hypothesis testing when the sample size is small and the population standard deviation is unknown. It accommodates additional uncertainty by being more spread out (having heavier tails) compared to the normal distribution. The t-distribution is characterized by its degrees of freedom, which depend on the sample size.

p-Value

The p-value is the probability, under the null hypothesis, of observing a test statistic as extreme as, or more extreme than, the one computed from the sample data. It quantifies the evidence against the null hypothesis, with smaller p-values indicating stronger evidence to reject it.

Significance Level

The significance level, often denoted by ? (for example, 0.05), is a threshold chosen by the researcher which defines how extreme the data must be to reject the null hypothesis. If the p-value is less than or equal to the significance level, the null hypothesis is rejected, suggesting that the results are statistically significant.

Transcript

00:01 So in this problem, we're given the following information.

00:03 That our null hypothesis includes a mu equaling 18.

00:06 Our alternative hypothesis is that mu does not equal 18.

00:10 We also are given that we have a sample size of 48, a sample mean of 17, and a sample standard deviation of 4 .5.

00:18 So the first thing the question asks us to do is to find a test statistic.

00:22 Now notice that we are given a sample standard deviation, not a population standard deviation.

00:28 So we have to find a t test statistic.

00:33 So here's the formula for a t test statistic.

00:35 It is equal to the sample mean minus the population mean over over the sample standard deviation divided by the square root of n.

00:47 Okay.

00:48 And for us, that is equal to 17 minus 18 over 4 .5 divided by the square root of 48, which is equal to negative 1 .54.

01:05 So this is our answer to part a.

01:08 Now, in the next question, we are asked to find a p value associated with negative 1 .54 and see whether we should accept or reject the null hypothesis with our alpha of 0 .05.

01:25 Okay.

01:26 So how do we find a p value for this? if we look at a t distribution, okay, we see that here, t equals zero, because this is a negative, this is a negative t test statistic.

01:43 Our t score is going to be to the left of t equals zero, somewhere out there.

01:48 So we are going to find the area to the left of t equals negative 1 .54...

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