A politician claims that he is supported by a clear majority of voters. In a recent survey, 45 out of 70 randomly selected voters indicated that they would vote for the politician. Calculate the value of test statistic



To test whether the politician is supported by a clear majority of voters, we need to conduct a hypothesis test for a proportion.

### Step 1: Set Up the Hypotheses
- **Null hypothesis (\( H_0 \))**: The proportion of voters who support the politician is 50%, or \( p = 0.5 \).
- **Alternative hypothesis (\( H_1 \))**: The proportion of voters who support the politician is greater than 50%, or \( p > 0.5 \).

### Step 2: Calculate the Test Statistic
The test statistic for a proportion is given by:

\[
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}
\]

Where:
- \( \hat{p} \) is the sample proportion.
- \( p_0 \) is the hypothesized population proportion.
- \( n \) is the sample size.

### Step 3: Plug in the Values
- Sample proportion, \( \hat{p} = \frac{45}{70} = 0.6429 \)
- Hypothesized proportion, \( p_0 = 0.5 \)
- Sample size, \( n = 70 \)

Now calculate the test statistic:

\[
z = \frac{0.6429 - 0.5}{\sqrt{\frac{0.5(1 - 0.5)}{70}}}
\]

\[
z = \frac{0.1429}{\sqrt{\frac{0.25}{70}}}
\]

\[
z = \frac{0.1429}{\sqrt{0.003571}}
\]

\[
z = \frac{0.1429}{0.0597} \approx 2.39
\]

The value of the test statistic is approximately 2.39.

Would you like more details or have any questions? 

Here are five related questions to further explore:
1. What does a z-score of 2.39 imply in this context?
2. How would you interpret the p-value associated with this test statistic?
3. What would the decision be at a significance level of 0.05?
4. How would the result change if the sample size were larger?
5. What if the survey showed 35 out of 70 in favor of the politician?

**Tip:** Always check whether the sample size is large enough for the normal approximation to be valid in hypothesis tests for proportions (usually \( np_0 \) and \( n(1 - p_0) \) should both be at least 5).

A politician claims that he is supported by a clear majority of voters. In a recent survey, 45 out of 70 randomly selected voters indicated that they would vote for the politician. Calculate the value of test statistic.

Learn how to calculate the test statistic in a hypothesis test for a proportion using real survey data. This example involves 45 out of 70 voters supporting a politician, with a detailed step-by-step solution showing how to apply statistical concepts and formulas.

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