Final answer:
To determine whether the sample of 150 students provides enough evidence to support the administration's belief that 70% of students are in favor of the cafe, we must conduct a hypothesis test for proportions. The sample proportion of 0.73 must be compared to the null hypothesis of 0.70 using a standard error calculation and a z-test to draw a conclusion.
Step-by-step explanation:
The Stats University administration's belief that 70% of students are in favor of the new cafe can be tested using a hypothesis test for proportions. With a sample of 150 students, we find that 110 support the cafe, which is a sample proportion (π) of 110/150 = 0.73. To determine if this sample provides enough evidence to support the administration's belief, we conduct a hypothesis test.
The null hypothesis (H0) would be that the true proportion of students who support the cafe is 0.70 (π = 0.70). The alternative hypothesis (Ha) could be that the true proportion is different from 0.70. We calculate the standard error of the sample proportion using the formula SE = √[p(1-p)/n], where p is the assumed population proportion (in this case, 0.70), and n is the sample size. The test statistic (z) is calculated as (sample proportion - population proportion) / SE. This z-value is then compared against a critical value from the z-table corresponding to the chosen significance level (commonly α = 0.05 for a two-tailed test).
Without carrying out the complete calculations and assuming the normality conditions are met, if the calculated z-value is less than the critical value, we would not reject the null hypothesis, implying the sample does not provide enough evidence against the administration's belief. Otherwise, if the z-value is higher, the sample would provide sufficient evidence to doubt the belief that 70% of students favor the cafe. However, to make a definitive conclusion, one must conduct the exact calculations and compare against the critical z-value.