Answer:
Based on the given data and the hypothesis test described, the answer is that we do not have sufficient evidence at the 0.05 level of significance to conclude that less than 30% of shoppers at Quincy Mall favor longer hours.
Explanation:
To determine whether the sample provides sufficient evidence to conclude that less than 30% of shoppers at Quincy Mall favor longer hours, we can conduct a hypothesis test using the following steps:
Step 1: State the null hypothesis and the alternative hypothesis.
Null hypothesis (H0): p >= 0.30 (more than or equal to 30% of shoppers favor longer hours)
Alternative hypothesis (Ha): p < 0.30 (less than 30% of shoppers favor longer hours)
Step 2: Determine the level of significance (alpha).
alpha = 0.05
Step 3: Calculate the test statistic and the p-value.
The test statistic can be calculated using the formula: z = (p - P) / sqrt(P(1-P) / n), where p is the sample proportion, P is the hypothesized proportion under the null hypothesis, and n is the sample size.
Using the given data, we have p = 125/450 = 0.278, P = 0.30, and n = 450.
Substituting these values into the formula, we get: z = (0.278 - 0.30) / sqrt(0.3 * 0.7 / 450) = -1.42
The p-value can be calculated using a standard normal distribution table or a calculator. The p-value for a one-tailed test with z = -1.42 is approximately 0.077.
Step 4: Compare the p-value with the level of significance and make a decision.
Since the p-value (0.077) is greater than the level of significance (0.05), we fail to reject the null hypothesis.
Therefore, we do not have sufficient evidence at the 0.05 level of significance to conclude that less than 30% of shoppers at Quincy Mall favor longer hours.
In other words, we cannot reject the possibility that 30% or more of shoppers at Quincy Mall favor longer hours based on the given sample data.