91Ó°ÊÓ

Please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test? (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. What is the value of the sample test statistic? (c) Find (or estimate) the \(P\) -value. Sketch the sampling distribution and show the area corresponding to the \(P\) -value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \(\alpha\) ? (e) State your conclusion in the context of the application. Let \(x\) be a random variable representing dividend yield of Australian bank stocks. We may assume that \(x\) has a normal distribution with \(\sigma=2.4 \%\). A random sample of 10 Australian bank stocks gave the following yields. \(\begin{array}{llllllllll}5.7 & 4.8 & 6.0 & 4.9 & 4.0 & 3.4 & 6.5 & 7.1 & 5.3 & 6.1\end{array}\) The sample mean is \(\bar{x}=5.38 \%\). For the entire Australian stock market, the mean dividend yield is \(\mu=4.7 \%\) (Reference: Forbes). Do these data indicate that the dividend yield of all Australian bank stocks is higher than \(4.7 \%\) ? Use \(\alpha=0.01\).

Step by step solution

01

Identify the Level of Significance and Formulate Hypotheses

The level of significance given in this problem is \( \alpha = 0.01 \). We need to test if the dividend yield of all Australian bank stocks is higher than 4.7%. Therefore, the null hypothesis \( H_0 \) will be that the mean dividend yield \( \mu \) equals 4.7%, i.e., \( H_0: \mu = 4.7 \). The alternative hypothesis \( H_a \) will be that the mean dividend yield \( \mu \) is greater than 4.7%, i.e., \( H_a: \mu > 4.7 \). Since we are testing if the mean is greater than a certain value, this will be a right-tailed test.

02

Choose the Sampling Distribution and Calculate Test Statistic

Given that the population standard deviation \( \sigma \) is known and the sample size is less than 30, we will use the standard normal distribution (\( Z \)-distribution) for the test. The test statistic \( Z \) is calculated using the formula: \[Z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}\]Where \( \bar{x} = 5.38 \), \( \mu = 4.7 \), \( \sigma = 2.4 \), and \( n = 10 \). Substituting these values:\[Z = \frac{5.38 - 4.7}{\frac{2.4}{\sqrt{10}}} \approx 0.91\]

03

Determine the P-Value

The \( P \)-value is the probability that the observed result, or something more extreme, would occur if the null hypothesis were true. For a right-tailed test with \( Z = 0.91 \), we look up this value in a standard normal distribution table or use a calculator to find \( P(Z > 0.91) \), which approximates to 0.1814.

04

Make a Decision Regarding the Null Hypothesis

To determine whether to reject the null hypothesis, we compare the \( P \)-value to \( \alpha \). Since \( P \)-value \( = 0.1814 \) is greater than \( \alpha = 0.01 \), we fail to reject the null hypothesis. This means there is not enough statistical evidence to support that the dividend yield of all Australian bank stocks is greater than 4.7% at the \( 0.01 \) significance level.

05

State the Conclusion

In conclusion, based on the sample data and the \( P \)-value obtained, we do not have sufficient evidence to conclude that the average dividend yield of Australian bank stocks is greater than 4.7% at the 1% significance level. Therefore, we fail to reject the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Level of Significance

When conducting a hypothesis test, the level of significance, denoted by \( \alpha \), is a key component to understand. It represents the threshold or cut-off point for determining whether the observed data is statistically significant. In simpler terms, it is the probability of rejecting a true null hypothesis, known as a Type I error.
A common choice for \( \alpha \) is 0.05, but in the given problem, it is specified as 0.01. This means the risk of mistakenly concluding that the average dividend yield of Australian bank stocks is above 4.7% is set at 1%. In practice, a lower \( \alpha \) such as 0.01 necessitates stronger evidence to reject the null hypothesis.
Choosing the right level of significance is crucial and depends on the context of the study, balancing the costs of errors against the need for evidence.

Sampling Distribution

Sampling distribution is a fundamental concept in hypothesis testing. When you take a random sample from a population, the statistics (like the mean or standard deviation) you calculate from this sample form their own distribution, known as the sampling distribution.
In our example, even though the sample size is small (n = 10), the population standard deviation \( \sigma \) is known. Therefore, it is appropriate to use the standard normal distribution, or \( Z \)-distribution, to test our hypothesis. The key idea is that while individual samples may vary, they follow a predictable distribution pattern when considered collectively.
Using a Z-distribution is particularly valid here as it helps quantify how far the sample mean of 5.38% is from the hypothesized population mean of 4.7%, accounting for sample size and variability. This choice makes the interpretation of results straightforward and mathematically robust.

P-Value

The \( P \)-value is a crucial concept in determining the result of a hypothesis test. It represents the probability of obtaining test results at least as extreme as the observed results, under the assumption that the null hypothesis is true. In our problem, the right-tailed \( P \)-value was found to be approximately 0.1814.
Calculating the \( P \)-value helps provide an objective criterion to determine whether the observed sample provides enough evidence against the null hypothesis. If the \( P \)-value is less than the pre-set level of significance \( \alpha \), we reject the null hypothesis.
Here, because the \( P \)-value of 0.1814 is greater than \( \alpha = 0.01 \), it indicates that the observed sample mean could reasonably occur under the null hypothesis, implying insufficient evidence to claim a difference in mean yield.

Null Hypothesis

Every hypothesis test starts with a null hypothesis, \( H_0 \), which represents a statement of no effect or no difference. It is what researchers typically try to disprove or discredit.
In the context of our problem, the null hypothesis is that the mean dividend yield \( \mu \) equals 4.7%, expressed mathematically as \( H_0: \mu = 4.7 \). This hypothesis posits that the observed yields of the sample do not significantly differ from the larger Australian stock market average.
Rejecting the null hypothesis would imply enough evidence to suggest that the true average differs from 4.7%, whereas failing to reject it, as we conclude in this case, suggests that there isn't enough statistical proof to claim a difference. Proper formulation and understanding of the null hypothesis are essential in the process of scientific inquiry and decision making.