Question

A certain party got 16.0 %of the votes in the last election. A new poll has been conducted,where 1000 randomly chosen people expressed which party they support. Out of them 170 said they support this party.

(a) Calculate a confidence interval for the current support of this party in the population,using the confidence level 95 %. Use the normal approximation. For the standard deviation, use an estimate that is calculated from the sample proportion 0.17. (Do not use the conservative estimate that is calculated from the proportion

(b) At significance level (alpha)α= 0.05, test the null hypothesis that the support of the party is still the same as in the last election, against the alternative hypothesis that the support has changed. Use the normal approximation. Calculate the p-value and express whether the null hypothesis is rejected or not

Answer 1

Given:

Sample size, n = 1000

Number of success, X = 170

Sample proportion, = X/n = 170/1000 = 0.17

Significance level, = 0.05

At 0.05 significance level, the critical value of Z is

Z/2 = Z0.05/2 = 1.96

95% confidence interval is

CI =   Z/2 × √(1-)/n

= 0.17  1.96 × √0.17(1-0.17)/1000

= 0.17 0.0233

= ( 0.1467, 0.1933)

Which is the required 95% conference interval.

2) Claimed proportion of votes in last election is p = 16% = 0.16

Hypothesis test:

Tje null and alternative hypothesis is

H0 : p = 0.16

Ha : p 0.16(This is a two tailed test)

Test statistics is

z = - p/√p(1-p)/n

= 0.17 - 0.16/√0.16(1-0.16)/1000

= 0.86

Test statistics is z = 0.86

p-value for two tailed test:

p-value = 2 × P(z>0.86)

= 2 × 0.1949 ....(from z-table)

= 0.3898

p-value = 0.3898

Since p-value is greater than significance level 0.05, we fail to reject null hypothesis.

Decision: Fail to reject null hypothesis, H0

Conclusion: There is insufficient evidence to conclude that the support of the party is still the same as in the last election.