Question

A well-known brokerage firm executive claimed that 50% of investors are currently confident of meeting their investment goals. An XYZ Investor Optimism Survey, conducted over a two week period, found that in a sample of 200 people, 41% of them said they are confident of meeting their goals. Test the claim that the proportion of people who are confident is smaller than 50% at the 0.05 significance level. The null and alternative hypothesis would be: H 0 : p ≤ 0.5 H 0 : p ≤ 0.5 H 1 : p > 0.5 H 1 : p > 0.5 H 0 : μ = 0.5 H 0 : μ = 0.5 H 1 : μ ≠ 0.5 H 1 : μ ≠ 0.5 H 0 : p ≥ 0.5 H 0 : p ≥ 0.5 H 1 : p < 0.5 H 1 : p < 0.5 H 0 : μ ≤ 0.5 H 0 : μ ≤ 0.5 H 1 : μ > 0.5 H 1 : μ > 0.5 H 0 : μ ≥ 0.5 H 0 : μ ≥ 0.5 H 1 : μ < 0.5 H 1 : μ < 0.5 H 0 : p = 0.5 H 0 : p = 0.5 H 1 : p ≠ 0.5 H 1 : p ≠ 0.5 The test is: right-tailed left-tailed two-tailed The test statistic is: (to 3 decimals) The p-value is: (to 4 decimals) Based on this we: Reject the null hypothesis Fail to reject the null hypothesis

Answer 1

Solution :

This hypothesis test is a one tailed test .

The null and alternative hypothesis is

H₀ : p 0.50

H_a : p < 0.50

= 0.41

Test statistic = z

= - P₀ / [P_{0 *} (1 - P₀ ) / n]

= 0.41 - 0.50 / [(0.50 * 0.50) / 200]

= -2.546

P-value = 0.0055

= 0.05

P-value <

Reject the null hypothesis .