Question

A local news agency would like to test the claim of the mayor that more than half the voters in the city support the construction. If p is the proportion of voters in favor of the construction, the news agency will test H 0 : p = .5 against H 1 : p < .5 to a .05 level of significance by surveying 500 voters. If 225 voters are in favor of the construction give the P-value and your conclusion.

Answer choices:

P=.0258, At least half the voters support construction.

P=.0258, Less than half the voters support construction.

P=.0129, Less than half the voters support construction.

P=.0129, At least half the voters support construction.

Answer 1

Solution:

Hypothesis are

H 0 : p = .5 against H 1 : p < .5

n = 500

x = 225

Let  $url=https://latex.codecogs.com/gif.latex?%5Chat%20p&t=1596167392&ymreqid=f4764063-0d40-e2b0-1cde-f0008d012f00&sig=sTtj6fUQ557EVceE5O50Rw--~D$ be the sample proportion.

$url=https://latex.codecogs.com/gif.latex?%5Chat%20p&t=1596167392&ymreqid=f4764063-0d40-e2b0-1cde-f0008d012f00&sig=sTtj6fUQ557EVceE5O50Rw--~D$ = x/n = 225/500 = 0.45

The test statistic z is

z =   $url=https://latex.codecogs.com/gif.latex?%5Cfrac%7B%5Chat%20p-p%7D%7B%5Csqrt%7B%5Cfrac%7Bp%281-p%29%7D%7Bn%7D%7D%7D&t=1596167392&ymreqid=f4764063-0d40-e2b0-1cde-f0008d012f00&sig=3WpFpTtEmFdUVRm2kKWcfg--~D$ =  (0.45 - 0.5)/$\sqrt{}$[0.5*(1 - 0.5)/500] = -2.23

Now , observe that there is < sign in H1

So , test is LEFT TAILED.

For left tailed test ,

p value = P[Z < z] = P[Z < -2.23] = 0.0129

Given , significance level $\alpha$ = 0.05

p value is less than $\alpha$

Reject H₀

We can conclude that there is sufficient evidence to support the claim that  Less than half the voters support construction.

So, correct option is

P=.0129, Less than half the voters support construction.