Question

4) A random sample of 42 observations was made on a variable XX. The sample mean was found to be 7.37 and the sample variance was found to be 5.53.

1. Construct a 99% confidence interval for the expected value (μμ) of XX.

2. Interpret the confidence interval.

3. Test the null hypothesis H0:μ=14.5H0:μ=14.5 against the alternative hypothesis H1:μ>14.5H1:μ>14.5 using a 10% significance level.

4. State any assumptions that you needed to make in order for your answers to parts a. and c. to be valid.

Answer 1

1. (3 marks) Construct a 99% confidence interval for the expected value (μμ) of XX.

The 99% confidence interval will be:

= x ± t*(√s2/n)

= 7.37 ± 2.70*(√5.53/42)

= (6.3899, 8.3501)

1. (2 marks) Interpret the confidence interval.

We are 99% confident that the true expected value (μ) of X is between 6.3899 and 8.3501.

1. (3 marks) Test the null hypothesis H0:μ=14.5H0:μ=14.5 against the alternative hypothesis H1:μ>14.5H1:μ>14.5 using a 10% significance level.

The test statistic, t = (x - µ)/s/√n

t = (7.37 - 14.5)/√5.53/42 = -19.65

The p-value is 1.0000.

Since the p-value (1.0000) is greater than the significance level (0.10), we fail to reject the null hypothesis.

Therefore, we cannot conclude that μ>14.5.

1. (2 marks) State any assumptions that you needed to make in order for your answers to parts a. and c. to be valid.

The one-sample t-test has four main assumptions:

• • The dependent variable must be continuous (interval/ratio).
• • The observations are independent of one another.
• • The dependent variable should be approximately normally distributed.
• • The dependent variable should not contain any outliers.

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