Question

Fifty percent of registered voters in a congressional district are registered Democrats. The Republican candidate takes a poll to assess his chances in a two-candidate race. He polls 1200 potential voters and finds that 621 plan to vote for the Republican candidate. Does the Republican candidate have a chance to win? Use a=0.05 PLEASE SHOW CLASSICAL & P-VALUE APPROACHES & TI84 METHODS FOR FINDING CV, P-VALUE & TEST STAT.

Answer 1

We use one proportion z test to test the claim that the republican candidate have a chance to win.

GIVEN:

Total number of potential voters

Voters who plan to vote for the republican candidate

HYPOTHESIS:

(That is, the republican candidate have a significant chance to win.)

(That is, the republican candidate does not have a significant chance to win.)

LEVEL OF SIGNIFICANCE:

TEST STATISTIC:

Since this is quite large sample we can use the central limit theorem to say that the distribution of proportions is approximately normal. Thus the test statistic is,

which follows standard normal distribution.

where

is the sample proportion

is the hypothesized value 0.5

CALCULATION:

Sample proportion

  

Now

  

  

  

Thus the calculated z test statistic value is .

CRITICAL VALUE:

The one tailed z critical value at is .

DECISION RULE:

.

INFERENCE:

Since the calculated z statistic value (1.215) is less than the z critical value (1.645), we fail to reject null hypothesis and conclude that the republican candidate have a significant chance to win since the proportion of voters who vote democrat will be in the minority ().

P VALUE:

The p value is given by,

From the z table, the p value is the value corresponding to 1.2 row and 0.02 column.

DECISION RULE FOR P VALUE:

INFERENCE:

Since the calculated P-value (0.8888) is greater than the significance level (0.05), we fail to reject null hypothesis and conclude that the republican candidate have a significant chance to win since the proportion of voters who vote democrat will be in the minority ().

Z TABLE: