Question Type 15: Performing and entire statistical test Exercises

In a study of 100 individuals, 50 smokers and 50 non-smokers are classified by the presence or absence of a respiratory condition. The results are recorded as follows:

• Smokers: 20 with the condition, 30 without the condition.
• Non-smokers: 10 with the condition, 40 without the condition.

At the $5\%$ significance level ($\alpha = 0.05$), perform a $\chi^2$ test of independence to determine whether there is an association between smoking status and the respiratory condition. State the null and alternative hypotheses, show the calculation of the test statistic, and state your conclusion.

A machine claims to fill bottles with a mean volume of $100\text{ mL}$. A sample of $15$ bottles is taken, yielding a sample mean volume of $103\text{ mL}$ and a sample standard deviation of $s = 5\text{ mL}$.

At the $\alpha = 0.05$ significance level, perform a one-tailed test to determine whether the true mean volume exceeds $100\text{ mL}$.

A sample of 10 measurements ($47, 48, 49, 48, 49, 50, 51, 52, 53, 54$) is taken. At the $5\%$ significance level, test whether the population mean is $50$ using a one-sample t-test.

A survey of 90 individuals was conducted to record their education level (High school, Bachelor, Master) and income bracket (Low, Mid, High). The observed counts are summarized in the following contingency table:

At the $\alpha = 0.01$ significance level, perform a $\chi^2$ test to determine whether education level and income bracket are independent.

Two classes' exam scores are compared. Class 1 ($n_1=20$) has $\bar{x}_1=78$ and $s_1=10$. Class 2 ($n_2=22$) has $\bar{x}_2=72$ and $s_2=12$.

At the $\alpha=0.05$ significance level, perform a pooled two-sample $t$-test to determine if there is a significant difference between the population mean scores of the two classes.

Group A ($n=8$) and Group B ($n=9$) have the following measurements:

Group A: $9, 12, 8, 11, 10, 9, 11, 13$

Group B: $7, 10, 6, 9, 8, 7, 11, 10, 8$

Use a pooled-variance two-sample $t$-test at the $5\%$ significance level to determine if there is a significant difference between the population means of the two groups.

A six-sided die is rolled 60 times with observed face counts: 4, 8, 9, 7, 12, 20. At the 5% significance level, perform a complete chi-squared goodness-of-fit test to determine if the die is fair.

A botanist classifies 200 plants into red, blue and green flowers with observed counts 95, 70 and 35 respectively. The expected proportions for these colors are 0.50, 0.30 and 0.20 respectively. At the $1\%$ level of significance, perform a chi-squared goodness-of-fit test for these proportions.

A sample of 12 observations yields $\bar{x} = 2.5$ and $s = 0.7$. Test at the $\alpha = 0.01$ significance level whether the population mean is 3.

A quality inspector counts the number of defects per item in a sample of 100 items, with the following results:

• 0 defects: 60 items
• 1 defect: 25 items
• 2 defects: 10 items
• 3 or more defects: 5 items

Assuming the data follows a Poisson distribution, perform a chi‐squared goodness‐of‐fit test at the $5\%$ level of significance, estimating the necessary parameter from the data.

Thirty people are classified by hair color (blonde, brown, black) and eye color (blue, brown, green) with the following observed counts:

• Blonde: 5 Blue, 3 Brown, 2 Green
• Brown: 3 Blue, 7 Brown, 0 Green
• Black: 2 Blue, 4 Brown, 4 Green

At a 5% significance level, test whether hair color and eye color are independent.

Sample 1 ($n_1 = 12$): $\bar{x}_1 = 5, s_1 = 1$; Sample 2 ($n_2 = 15$): $\bar{x}_2 = 6.2, s_2 = 1.5$.

Using a Welch two-sample $t$-test at $\alpha = 0.05$ (one-tailed, $H_1: \mu_1 < \mu_2$), test the difference between the population means.