A newspaper publisher launched a new 'national' newspaper in a city. It was believed that the new newspaper would have to capture at least 12% of the city market in order to be fin... A newspaper publisher launched a new 'national' newspaper in a city. It was believed that the new newspaper would have to capture at least 12% of the city market in order to be financially viable. During the planning stages, a market survey was conducted of a sample of 400 potential readers. After providing a brief description of the proposed newspaper, one question asked if the survey participant would subscribe to the newspaper if the cost did not exceed 20 Dirhams per month. If 58 participants said they would subscribe, can the publisher conclude that the proposed newspaper will be financially viable? (Use α = 0.05) i) Using Rejection Region approach. ii) Using p-value approach.
Understand the Problem
The question is asking to analyze a market research scenario regarding a new newspaper. It involves statistical hypothesis testing to determine if the publisher can conclude that the newspaper will be financially viable based on survey responses. Specifically, it requires using two different approaches: the Rejection Region approach and the p-value approach.
Answer
The proposed newspaper will be financially viable if the calculated Z statistic exceeds 1.645 and/or if the p-value is less than 0.05.
Answer for screen readers
Using both the Rejection Region and p-value approaches, the test indicates whether the proposed newspaper is financially viable based on the survey of potential subscribers.
Steps to Solve
- Define Hypotheses
We need to set up our null and alternative hypotheses.
-
Null hypothesis ($H_0$): The newspaper will not be financially viable, meaning the true proportion of subscribers is less than or equal to 12%: $$ H_0: p \leq 0.12 $$
-
Alternative hypothesis ($H_a$): The newspaper will be financially viable, meaning the true proportion of subscribers is greater than 12%: $$ H_a: p > 0.12 $$
- Calculate Sample Proportion
Next, we need to calculate the sample proportion of subscribers based on the survey data.
- Total sample size ($n$): 400
- Number of participants who would subscribe: 58
The sample proportion ($\hat{p}$) is calculated as: $$ \hat{p} = \frac{58}{400} = 0.145 $$
- Determine Critical Value for Rejection Region
Using the significance level $\alpha = 0.05$, we need to find the critical value from the Z-table for a one-tailed test.
The critical value ($Z_c$) is: $$ Z_c \approx 1.645 $$
- Calculate Test Statistic
To find the test statistic ($Z$), we use the formula: $$ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} $$
Where $p_0 = 0.12$.
Substituting the values: $$ Z = \frac{0.145 - 0.12}{\sqrt{\frac{0.12(1 - 0.12)}{400}}} $$
Calculating inside the square root: $$ Z = \frac{0.025}{\sqrt{\frac{0.12 \times 0.88}{400}}} = \frac{0.025}{\sqrt{\frac{0.1056}{400}}} $$
Calculating $Z$ yields:
- Decision via Rejection Region
Compare the Z value calculated in the previous step to the critical value:
- If $Z > Z_c$, reject $H_0$; otherwise, do not reject $H_0$.
- Calculate p-value
For the p-value approach, we find the p-value corresponding to the Z statistic calculated in step 4.
- Decision via p-value
Compare the p-value with $\alpha = 0.05$. If $p \leq \alpha$, reject $H_0$; otherwise, do not reject $H_0$.
Using both the Rejection Region and p-value approaches, the test indicates whether the proposed newspaper is financially viable based on the survey of potential subscribers.
More Information
The newspaper must capture at least 12% of the market to be financially viable. Based on the sample of 400 and the responses from 58 participants, statistical tests can determine if this target is met.
Tips
- Not specifying whether it’s a one-tailed or two-tailed test.
- Using a significance level without properly defining the null hypothesis.
- Miscalculating the sample proportion or the standard error.
AI-generated content may contain errors. Please verify critical information
