Virtual call centers are staffed by individuals working out of their homes. ABC Services is considering employing home-based agents, but only if a level of customer satisfaction greater than or equal to 80% can be maintained. A trial program was conducted using home-based agents. In an SRS of 300 customers, 252 reported that they were satisfied with the service they received.
(a) What is sample proportion ����?
(b) What is the standard error of ����?
(c) Compute a 95% Confidence Interval for the true trial program satisfaction level.
(d) Conduct a hypothesis test to determine if the trial program sample supports the conclusion that the home-based agent satisfaction level meets the level required by ABC Services. Be sure to state your Null and Alternative Hypotheses, test statistic, p-value, and conclusion.
(a) The sample proportion is the ratio of the number of customers satisfied to the total number of customers surveyed. In this case, the sample proportion is calculated by dividing the number of satisfied customers (252) by the total number of customers surveyed (300).
So, the sample proportion is 252/300 = 0.84.
(b) The standard error of the sample proportion can be calculated using the formula:
SE = sqrt((p*(1-p)) / n)
where p is the sample proportion and n is the sample size.
In this case, p = 0.84 and n = 300.
SE = sqrt((0.84*(1-0.84)) / 300)
SE = sqrt((0.84*0.16) / 300)
SE = sqrt(0.1344 / 300)
SE = sqrt(0.000448)
SE = 0.021
Therefore, the standard error of the sample proportion is 0.021.
(c) To compute a 95% Confidence Interval for the true trial program satisfaction level, we can use the formula:
CI = sample proportion ± (critical value * standard error)
The critical value is based on the desired confidence level and can be obtained from a standard normal distribution table. For a 95% confidence level, the critical value is approximately 1.96.
CI = 0.84 ± (1.96 * 0.021)
CI = 0.84 ± 0.041
Therefore, the 95% Confidence Interval for the true trial program satisfaction level is (0.799, 0.881).
(d) To conduct a hypothesis test, we can set up the following Null and Alternative Hypotheses:
Null Hypothesis (H0): The home-based agent satisfaction level is less than 80% (p < 0.80)
Alternative Hypothesis (Ha): The home-based agent satisfaction level is greater than or equal to 80% (p ≥ 0.80)
We can use a one-sided z-test to test these hypotheses. The test statistic can be calculated using the formula:
z = (p - P0) / sqrt((P0 * (1 - P0)) / n)
where p is the sample proportion, P0 is the hypothesized proportion under the null hypothesis (0.80 in this case), and n is the sample size.
In this case, p = 0.84, P0 = 0.80, and n = 300.
z = (0.84 - 0.80) / sqrt((0.80 * (1 - 0.80)) / 300)
z = 0.04 / sqrt(0.16/300)
z = 0.04 / sqrt(0.000533)
z = 0.04 / 0.023
z ≈ 1.74
To find the p-value, we can compare the test statistic to the standard normal distribution. The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated. From the standard normal distribution table, the p-value is approximately 0.041.
Since the p-value (0.041) is less than the significance level (α = 0.05), we reject the null hypothesis. Therefore, the trial program sample supports the conclusion that the home-based agent satisfaction level meets the level required by ABC Services.
Conclusion: Based on the hypothesis test, there is enough evidence to support the conclusion that the home-based agent satisfaction level meets or exceeds the level required by ABC Services.