can you show me the caculation :)
Last year, 50% of MNM, Inc. employees were female. It is believed that there has been a reduction in the percentage of females in the company. This year, in a random sample of 400 employees, 180 were female.
a.Give the null and the alternative hypotheses.
b.At 95% confidence using the critical value approach, determine if there has been a significant reduction in the proportion of females.
c.Show that the p-value approach results in the same conclusion as that of Part b.
a. The null hypothesis (H0) would be that the proportion of females remains the same as last year, i.e., p = 0.50. The alternative hypothesis (Ha) would be that there has been a reduction in the proportion of females, i.e., p < 0.50.
b. To determine if there has been a significant reduction in the proportion of females, we can perform a hypothesis test using the critical value approach.
1. Calculate the sample proportion of females (p̂) in the random sample of 400 employees: p̂ = 180/400 = 0.45
2. Calculate the standard error of the proportion using the formula: SE = sqrt((p̂(1 - p̂))/n), where n is the sample size. In this case, SE = sqrt((0.45(1 - 0.45))/400) ≈ 0.0224
3. Calculate the test statistic, which follows the standard normal distribution under the null hypothesis, using the formula: z = (p̂ - p) / SE, where p is the hypothesized proportion under the null hypothesis. In this case, z = (0.45 - 0.50) / 0.0224 ≈ -2.23
4. Determine the critical value at a 95% confidence level. For a two-tailed test, the critical value would be ±1.96.
5. Compare the test statistic to the critical value. Since -2.23 < -1.96, we reject the null hypothesis and conclude that there has been a significant reduction in the proportion of females at MNM, Inc.
c. To show that the p-value approach results in the same conclusion as part b, we can calculate the p-value for the test.
1. The p-value represents the probability of observing a test statistic as extreme as the one calculated (or more extreme) under the null hypothesis. In this case, we want to calculate the probability of observing a z-score of -2.23 or more extreme.
2. The p-value can be calculated using a standard normal distribution table or a statistical software. For a z-score of -2.23, the p-value is approximately 0.0131.
3. Compare the p-value to the significance level (α) set for the test. If the p-value is less than α, we reject the null hypothesis. In this case, with a significance level of 0.05, the p-value of 0.0131 is less than 0.05. Therefore, we reject the null hypothesis and conclude that there has been a significant reduction in the proportion of females at MNM, Inc.
Both the critical value approach and the p-value approach result in the same conclusion: there has been a significant reduction in the proportion of females at MNM, Inc.
a. The null hypothesis (H0) states that there has been no reduction in the proportion of females in the company. The alternative hypothesis (Ha) states that there has been a reduction in the proportion of females in the company.
b. To determine if there has been a significant reduction in the proportion of females, we will conduct a hypothesis test at a 95% confidence level using the critical value approach. The critical value for a two-tailed test at 95% confidence is approximately ±1.96.
Here's how you can perform the calculation:
1. Calculate the sample proportion of females:
Sample proportion (p̂) = number of females in the sample / sample size = 180 / 400 = 0.45
2. Calculate the standard error of the proportion:
Standard error (SE) = √((p̂*(1-p̂)) / n)
where n is the sample size.
SE = √((0.45*(1-0.45)) / 400) ≈ 0.025
3. Calculate the test statistic:
Test statistic (Z) = (p̂ - p) / SE
where p is the hypothesized proportion under the null hypothesis.
Here, p = 0.50 (percentage of females last year).
Z = (0.45 - 0.50) / 0.025 ≈ -2.00
4. Compare the test statistic with the critical value:
Since this is a two-tailed test, we need to compare the absolute value of the test statistic with the absolute value of the critical value.
|Z| = | -2.00 | = 2.00
The critical value for a two-tailed test at 95% confidence is ±1.96.
Since |Z| > |critical value| (2.00 > 1.96), we reject the null hypothesis.
Therefore, there is evidence to suggest that there has been a significant reduction in the proportion of females in the company.
c. To show that the p-value approach results in the same conclusion, we calculate the p-value using the test statistic obtained in Part b.
1. Determine the p-value:
The p-value is the probability of obtaining a test statistic as extreme as (or more extreme than) the one observed, assuming that the null hypothesis is true.
For a two-tailed test, the p-value is the probability of obtaining a test statistic as extreme as (or more extreme than) -2.00 or 2.00.
Using a standard normal distribution table or calculator, we find that the p-value is approximately 0.0455.
2. Compare the p-value with the significance level (α):
Since the p-value (0.0455) is less than the significance level (α = 0.05), we reject the null hypothesis.
Therefore, the p-value approach also leads us to conclude that there has been a significant reduction in the proportion of females in the company, which matches the conclusion obtained in Part b.