Math Problem Statement

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

Solution

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 (H0H_0): The proportion of voters who support the politician is 50%, or p=0.5p = 0.5.
  • Alternative hypothesis (H1H_1): The proportion of voters who support the politician is greater than 50%, or p>0.5p > 0.5.

Step 2: Calculate the Test Statistic

The test statistic for a proportion is given by:

z=p^p0p0(1p0)nz = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}

Where:

  • p^\hat{p} is the sample proportion.
  • p0p_0 is the hypothesized population proportion.
  • nn is the sample size.

Step 3: Plug in the Values

  • Sample proportion, p^=4570=0.6429\hat{p} = \frac{45}{70} = 0.6429
  • Hypothesized proportion, p0=0.5p_0 = 0.5
  • Sample size, n=70n = 70

Now calculate the test statistic:

z=0.64290.50.5(10.5)70z = \frac{0.6429 - 0.5}{\sqrt{\frac{0.5(1 - 0.5)}{70}}}

z=0.14290.2570z = \frac{0.1429}{\sqrt{\frac{0.25}{70}}}

z=0.14290.003571z = \frac{0.1429}{\sqrt{0.003571}}

z=0.14290.05972.39z = \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 np0np_0 and n(1p0)n(1 - p_0) should both be at least 5).

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Math Problem Analysis

Mathematical Concepts

Statistics
Hypothesis Testing
Proportion

Formulas

Test statistic for a proportion: z = (p̂ - p0) / √[p0(1 - p0) / n]

Theorems

Normal approximation to the binomial distribution in large samples

Suitable Grade Level

Undergraduate (Statistics 101)