two samples of data regarding salary of Co A and Co B determine if you can be 99% sure that Co?
A makes more than Co B? Company A - Mean 55,000, Std Dev 5,000, Sample Size 200; Co B mean 52,000, Std Dev 3,000 Sample size 400 please show steps thanks
To determine if you can be 99% confident that Company A makes more than Company B, we can use a hypothesis test.
Step 1: State the null and alternative hypotheses.
The null hypothesis (H0) assumes that there is no difference in the mean salaries of Company A and Company B.
H0: μ1 ≤ μ2
The alternative hypothesis (H1) assumes that Company A's mean salary is higher than Company B's mean salary.
H1: μ1 > μ2
Step 2: Set the significance level (α).
In this case, the significance level is given as 99%, which means that we want to be 99% confident in our conclusion. Therefore, α = 1 - 0.99 = 0.01.
Step 3: Determine the test statistic.
We can use the z-test statistic to compare the means of two independent samples. The formula for the z-test statistic is:
z = (x1 - x2 - 0) / sqrt((s1^2 / n1) + (s2^2 / n2))
where x1 and x2 are the sample means, s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.
Plugging in the given values:
x1 = 55000, s1 = 5000, n1 = 200
x2 = 52000, s2 = 3000, n2 = 400
z = (55000 - 52000 - 0) / sqrt((5000^2 / 200) + (3000^2 / 400))
Step 4: Calculate the critical value.
Since the alternative hypothesis is μ1 > μ2, we are interested in the right-tailed critical region. We can look up the critical z-value from the standard normal distribution table corresponding to our significance level (α = 0.01). In this case, the critical z-value is approximately 2.33.
Step 5: Compare the test statistic with the critical value.
If the test statistic is greater than the critical value, we reject the null hypothesis (H0).
In this case, if the calculated z-value is greater than 2.33, we reject the null hypothesis and conclude that there is sufficient evidence to suggest that Company A makes more than Company B, with a 99% confidence level.
Now, let's calculate the test statistic:
z = (55000 - 52000 - 0) / sqrt((5000^2 / 200) + (3000^2 / 400))
= 3000 / sqrt((25000000 / 200) + (9000000 / 400))
= 3000 / sqrt(125000 + 22500)
= 3000 / sqrt(147500)
= 3000 / 383.405
≈ 7.824
Since the calculated z-value (7.824) is greater than the critical value (2.33), we reject the null hypothesis. Therefore, we can be 99% confident that Company A makes more than Company B based on the given data.