Let x be a random variable representing dividend yield of Australian bank stocks. We may assume that x has a normal distribution with o=2.5%. A random sample of 24 Australian bank stocks has a sample mean of x=8.07%. For the entire Australian stock market, the mean dividend yield is u=6%. Do these data indicate that the dividend yield of all Australian bank stocks is higher than 6% ? Use a=0.1. Are the data statistically significant at the given level of significance? Based on your answers, will you reject or fail to reject the null hypothesis? The P-value is less than the level of significance and so the data are statistically significant. Thus, we reject the null hypothesis. The P-value is greater than the level of significance and so the data are not statistically significant. Thus, we fail to reject the null hypothesis. The P-value is less than the level of significance and so the data are not statistically significant. Thus, we fail to reject the null hypothesis. O The P-value is greater than the level of significance and so the data are statistically significant. Thus, we fail to reject the null hypothesis. O The P-value is less than the level of significance and so the data are statistically significant. Thus, we fail to reject the null hypothesis.
Let x be a random variable representing dividend yield of Australian bank stocks. We may assume that x has a normal distribution with o=2.5%. A random sample of 24 Australian bank stocks has a sample mean of x=8.07%. For the entire Australian stock market, the mean dividend yield is u=6%. Do these data indicate that the dividend yield of all Australian bank stocks is higher than 6% ? Use a=0.1. Are the data statistically significant at the given level of significance? Based on your answers, will you reject or fail to reject the null hypothesis? The P-value is less than the level of significance and so the data are statistically significant. Thus, we reject the null hypothesis. The P-value is greater than the level of significance and so the data are not statistically significant. Thus, we fail to reject the null hypothesis. The P-value is less than the level of significance and so the data are not statistically significant. Thus, we fail to reject the null hypothesis. O The P-value is greater than the level of significance and so the data are statistically significant. Thus, we fail to reject the null hypothesis. O The P-value is less than the level of significance and so the data are statistically significant. Thus, we fail to reject the null hypothesis.
MATLAB: An Introduction with Applications
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![### Hypothesis Testing of Dividend Yield of Australian Bank Stocks
**Problem Statement:**
Let \( x \) be a random variable representing the dividend yield of Australian bank stocks. We may assume that \( x \) has a normal distribution with \( \sigma = 2.5\% \). A random sample of 24 Australian bank stocks has a sample mean of \( \bar{x} = 8.07\% \). For the entire Australian stock market, the mean dividend yield is \( \mu = 6\% \). Do these data indicate that the dividend yield of all Australian bank stocks is higher than 6%? Use \( \alpha = 0.1 \). Are the data statistically significant at the given level of significance? Based on your answers, will you reject or fail to reject the null hypothesis?
**Options:**
1. ○ The P-value is less than the level of significance and so the data are statistically significant. Thus, we reject the null hypothesis.
2. ○ The P-value is greater than the level of significance and so the data are not statistically significant. Thus, we fail to reject the null hypothesis.
3. ○ The P-value is less than the level of significance and so the data are not statistically significant. Thus, we fail to reject the null hypothesis.
4. ○ The P-value is greater than the level of significance and so the data are statistically significant. Thus, we fail to reject the null hypothesis.
5. ○ The P-value is less than the level of significance and so the data are statistically significant. Thus, we fail to reject the null hypothesis.
---
### Explanation:
To determine if the dividend yield of all Australian bank stocks is higher than 6%:
1. **Formulate Hypotheses:**
- Null Hypothesis (\(H_0\)): \( \mu = 6\% \)
- Alternative Hypothesis (\(H_1\)): \( \mu > 6\% \)
2. **Determine the Test Statistic:**
- The test statistic for a sample mean when the population standard deviation (\(\sigma\)) is known is given by:
\[
z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}
\]
- Where:
- \(\bar{x} = 8.07\%\) (sample mean)
- \(\mu = 6\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6805edd-9acc-4ccd-9c7a-22df1558ea0b%2F93374e9e-1083-420b-b433-6beff3f19090%2Fj0o05r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Hypothesis Testing of Dividend Yield of Australian Bank Stocks
**Problem Statement:**
Let \( x \) be a random variable representing the dividend yield of Australian bank stocks. We may assume that \( x \) has a normal distribution with \( \sigma = 2.5\% \). A random sample of 24 Australian bank stocks has a sample mean of \( \bar{x} = 8.07\% \). For the entire Australian stock market, the mean dividend yield is \( \mu = 6\% \). Do these data indicate that the dividend yield of all Australian bank stocks is higher than 6%? Use \( \alpha = 0.1 \). Are the data statistically significant at the given level of significance? Based on your answers, will you reject or fail to reject the null hypothesis?
**Options:**
1. ○ The P-value is less than the level of significance and so the data are statistically significant. Thus, we reject the null hypothesis.
2. ○ The P-value is greater than the level of significance and so the data are not statistically significant. Thus, we fail to reject the null hypothesis.
3. ○ The P-value is less than the level of significance and so the data are not statistically significant. Thus, we fail to reject the null hypothesis.
4. ○ The P-value is greater than the level of significance and so the data are statistically significant. Thus, we fail to reject the null hypothesis.
5. ○ The P-value is less than the level of significance and so the data are statistically significant. Thus, we fail to reject the null hypothesis.
---
### Explanation:
To determine if the dividend yield of all Australian bank stocks is higher than 6%:
1. **Formulate Hypotheses:**
- Null Hypothesis (\(H_0\)): \( \mu = 6\% \)
- Alternative Hypothesis (\(H_1\)): \( \mu > 6\% \)
2. **Determine the Test Statistic:**
- The test statistic for a sample mean when the population standard deviation (\(\sigma\)) is known is given by:
\[
z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}
\]
- Where:
- \(\bar{x} = 8.07\%\) (sample mean)
- \(\mu = 6\
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