Questions: Use the results from a survey of a simple random sample of 1219 adults. Among the 1219 respondents, 63% rated themselves as above average drivers. We want to test the claim that 11/20 of adults rate themselves as above average drivers. Complete parts (a) through (c). a. Identify the actual number of respondents who rated themselves as above average drivers. 768 (Round to the nearest whole number as needed.) b. Identify the sample proportion and use the symbol that represents it. p̂=0.63 (Type an integer or a decimal rounded to two decimal places as needed.) c. For the hypothesis test, identify the value used for the population proportion and use the symbol that represents it. (Type an integer or a decimal rounded to two decimal places as needed.)

Use the results from a survey of a simple random sample of 1219 adults. Among the 1219 respondents, 63% rated themselves as above average drivers. We want to test the claim that 11/20 of adults rate themselves as above average drivers. Complete parts (a) through (c). a. Identify the actual number of respondents who rated themselves as above average drivers. 768 (Round to the nearest whole number as needed.) b. Identify the sample proportion and use the symbol that represents it. p̂=0.63 (Type an integer or a decimal rounded to two decimal places as needed.) c. For the hypothesis test, identify the value used for the population proportion and use the symbol that represents it. (Type an integer or a decimal rounded to two decimal places as needed.)
Transcript text: Use the results from a survey of a simple random sample of 1219 adults. Among the 1219 respondents, $63 \%$ rated themselves as above average drivers. We want to test the claim that $\frac{11}{20}$ of adults rate themselves as above average drivers. Complete parts (a) through (c). a. Identify the actual number of respondents who rated themselves as above average drivers. 768 (Round to the nearest whole number as needed.) b. Identify the sample proportion and use the symbol that represents it. \[ \hat{p}=0.63 \] (Type an integer or a decimal rounded to two decimal places as needed.) c. For the hypothesis test, identify the value used for the population proportion and use the symbol that represents it. (Type an integer or a decimal rounded to two decimal places as needed.)
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Solution

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Solution Steps

Step 1: Actual Number of Respondents

The actual number of respondents who rated themselves as above average drivers is calculated as follows:

\[ \text{Actual Number} = n \cdot \hat{p} = 1219 \cdot 0.63 = 768 \]

Step 2: Sample Proportion

The sample proportion, denoted as \( \hat{p} \), is given by:

\[ \hat{p} = 0.63 \]

Step 3: Population Proportion

The hypothesized population proportion, denoted as \( p_0 \), is calculated as:

\[ p_0 = \frac{11}{20} = 0.55 \]

Step 4: Hypothesis Test Calculation

To perform the hypothesis test, we calculate the Z-test statistic using the formula:

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

Substituting the values:

\[ Z = \frac{0.63 - 0.55}{\sqrt{\frac{0.55(1 - 0.55)}{1219}}} = 5.6144 \]

Step 5: P-value and Critical Region

The P-value associated with the test statistic is:

\[ \text{P-value} = 0.0 \]

The critical region for a two-tailed test at a significance level of \( \alpha = 0.05 \) is defined as:

\[ Z < -1.96 \quad \text{or} \quad Z > 1.96 \]

Final Answer

  • Actual number of respondents: \\(\boxed{768}\\)
  • Sample proportion: \\(\boxed{0.63}\\)
  • Population proportion: \\(\boxed{0.55}\\)
  • Test statistic: \\(\boxed{5.6144}\\)
  • P-value: \\(\boxed{0.0}\\)
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