Answer:
Step 1: Find the common denominator:
The common denominator for (2X + 1) and (X - 1) is (2X + 1)(X - 1).
Explanation:
Hypothesis Test: Mean Mark
User
Testing a mean when Population Variance is Unknown
1. The personnel department of a company has developed an aptitude test for screening potential employees. The person who devised the test asserted that the mean mark attained would be 100. The following results were obtained with random sample of applicants
X = 110,
s= 8,
n 13
Test the null hypothesis that the mean mark is equal to 100 against the alternative that the mean mark is greater than 100, at the 5% level of significance or 0.05?
ChatGPT
To test the null hypothesis that the mean mark is equal to 100 against the alternative that the mean mark is greater than 100, we can use a one-sample t-test since the population variance is unknown. Here's how you can perform the test:
Step 1: State the null and alternative hypotheses:
Null hypothesis (H₀): The mean mark is equal to 100.
Alternative hypothesis (H₁): The mean mark is greater than 100.
Step 2: Set the significance level (α):
In this case, the significance level is given as 0.05 or 5%.
Step 3: Compute the test statistic:
The test statistic for a one-sample t-test is calculated using the formula:
t = (X - μ) / (s / √n)
where X is the sample mean, μ is the population mean under the null hypothesis, s is the sample standard deviation, and n is the sample size.
Given:
X = 110 (sample mean)
s = 8 (sample standard deviation)
n = 13 (sample size)
μ (population mean under the null hypothesis) = 100
Substituting the values into the formula, we get:
t = (110 - 100) / (8 / √13)
t = 10 / (8 / √13)
t ≈ 3.012
Step 4: Determine the critical value:
Since the alternative hypothesis is one-tailed (greater than), we need to find the critical value for a one-tailed test at a 5% significance level with (n - 1) degrees of freedom. In this case, the degrees of freedom are 13 - 1 = 12.
Using a t-distribution table or statistical software, the critical value at α = 0.05 and 12 degrees of freedom is approximately 1.782.
Step 5: Make a decision:
If the test statistic t is greater than the critical value, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
In this case, the test statistic t is approximately 3.012, which is greater than the critical value of 1.782. Therefore, we reject the null hypothesis.
Step 6: State the conclusion:
Based on the sample data, there is sufficient evidence to support the claim that the mean mark is greater than 100.
User
X+3. X-2
___. - _____
2x+1. X-1
ChatGPT
To simplify the expression (X + 3)/(2X + 1) - (X - 2)/(X - 1), we need to find a common denominator and combine the fractions. Here's the step-by-step solution:
Step 1: Find the common denominator:
The common denominator for (2X + 1) and (X - 1) is (2X + 1)(X - 1).
Step 2: Rewrite the fractions with the common denominator:
(X + 3)/(2X + 1) - (X - 2)/(X - 1) = [(X + 3)(X - 1)]/[(2X + 1)(X - 1)] - [(X - 2)(2X + 1)]/[(2X + 1)(X - 1)]
Step 3: Combine the fractions:
[(X + 3)(X - 1)] - [(X - 2)(2X + 1)] / [(2X + 1)(X - 1)]
Expanding the expressions:
[X^2 + 3X - X - 3] - [2X^2 - 4X + X - 2] / [(2X + 1)(X - 1)]
Simplifying:
[X^2 + 2X - 3] - [2X^2 - 3X - 2] / [(2X + 1)(X - 1)]
Distribute the negative sign:
[X^2 + 2X - 3 - 2X^2 + 3X + 2] / [(2X + 1)(X - 1)]
Combine like terms:
(-X^2 + 5X - 1) / [(2X + 1)(X - 1)]
So, the simplified expression is (-X^2 + 5X - 1) / [(2X + 1)(X - 1)].