A manager questioned 100 employees selected as from a workforce of 7000 concerning the opinions of a proposed change in the company medical insurance plan.the medical insurance sales person claimed at least 80% of your workforce will favor this change.However,the manager's sample of 100 showed that only 70 employees in the sample favored the change.is the salespersons claim true? If not,what proportion of the total workforce favors the change?
To determine if the salesperson's claim is true and to find out the proportion of the total workforce that favors the change, we can use statistical inference.
First, let's define the null hypothesis (H0) and the alternative hypothesis (Ha):
- Null hypothesis: At least 80% of the total workforce favors the change.
- Alternative hypothesis: Less than 80% of the total workforce favors the change.
We are given a sample of 100 employees, out of a total workforce of 7000. In this sample, 70 employees favored the change.
To analyze this data and test the salesperson's claim, we can perform a hypothesis test using the binomial distribution. We assume that the opinions of the employees in the sample represent the opinions of the entire workforce.
We need to calculate the p-value, which represents the probability of observing a result as extreme as or more extreme than the sample if the null hypothesis is true.
In this case, we have a one-sample test, comparing a proportion to a hypothesized value. The test statistic can be calculated using the formula:
z = (p - P) / sqrt[(P * (1 - P)) / n]
where:
- p is the proportion of employees in the sample who favored the change (70/100 = 0.7),
- P is the hypothesized proportion of the total workforce who favors the change (0.8 in this case),
- n is the sample size (100).
Using this formula, we have:
z = (0.7 - 0.8) / sqrt[(0.8 * (1 - 0.8)) / 100]
z = -1 / sqrt[0.16 / 100]
z = -1 / 0.04
z = -25
Now, we can use the standard normal distribution (z-distribution) to find the p-value associated with the calculated z-value. Based on the z-value of -25, the p-value is almost 0, which is extremely small.
Since the p-value is very small, much smaller than the commonly chosen significance level of 0.05, we reject the null hypothesis.
Therefore, we conclude that the salesperson's claim is not true. The proportion of the total workforce that favors the change is less than 80%. However, we also need to calculate this proportion.
Based on the obtained sample proportion of 0.7, we estimate that approximately 70% of the total workforce favors the change.
To determine whether the salesperson's claim is true or not, we need to conduct a hypothesis test. Let's outline the steps to do so:
Step 1: Define the hypotheses
- Null Hypothesis (H0): At least 80% of the total workforce favors the change.
- Alternative Hypothesis (H1): Less than 80% of the total workforce favors the change.
Step 2: Set the significance level
Let's assume the significance level (α) is 0.05, which is a common choice.
Step 3: Calculate the test statistic
We need to use a one-sample proportion hypothesis test. We can calculate the test statistic using the following formula:
z = (p - P) / sqrt((P * (1-P)) / n)
where:
p = sample proportion
P = claimed proportion by the salesperson
n = sample size
In our case:
p = 70/100 = 0.7
P = 0.8
n = 100
z = (0.7 - 0.8) / sqrt((0.8 * (1-0.8)) / 100)
Step 4: Calculate the p-value
To find the p-value associated with the test statistic, we use a Z-table or a calculator. The p-value represents the probability of observing a test statistic as extreme as the one calculated if the null hypothesis is true.
Step 5: Compare the p-value with the significance level
If the p-value is less than the significance level (α), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Based on the given data, let's calculate the test statistic, p-value, and make the final conclusion.