Confidence level of 95% and conducting a one-sided hypothesis te

CFA Level I Quant
#1

A campaign strategist wants to determine whether demographic shifts have caused a drop in allegiance to the Uniformian Party in Bowie County. Historically, around 62% of the county’s registered voters have supported the Uniformians. In a survey of 196 registered voters, 57% indicated that they would vote for the Uniformians in the next election.

Assuming a confidence level of 95% and conducting a one-sided hypothesis test, which of the following should the strategist do?

Options:
Accept the hypothesis that the proportion of Uniformian voters has not changed.
Accept the hypothesis that the proportion of Uniformian voters has decreased.
Conclude that the proportion of Uniformian voters is now between 56% and 62%.
There is not enough evidence to support the hypothesis that the proportion of Uniformian voters has decreased.

#2

Null hypothesis: H_0 \geq 0.62

Alternative hypothesis: H_a < 0.62

The standard deviation of the sampling distribution:

\sigma = \sqrt {\frac{P(1-P)}{n}} = \sqrt {\frac{0.62(1-0.62)}{196}} = 0.03467

The Z test statistic:

Z = \frac{p - P}{\sigma} = \frac{0.57 - 0.62}{0.03467} = -1.4422

The critical Z based on 95% confidence level (5% significance level; left-tailed) is -1.65.

Since the test statistic (-1.44) is greater than the critical value (-1.65), we cannot reject the null hypothesis. The proportion of Uniformian voters has not decreased.