Q: A researcher is testing whether a new teaching method improves test scores. She collects a sample of 40 students and finds a sample mean of 78 with a known population standard deviation of 10. What is the 95% confidence interval for the population mean?

$$ 78 \pm 1.96 \times \frac{10}{\sqrt{40}} $$

Q: Which formula would you use to calculate the expected frequency for a cell in a chi-square test of independence?

$$ E = \frac{(\text{row total}) \times (\text{column total})}{\text{grand total}} $$

Question 1

A pharmaceutical company wants to compare the mean recovery times of two different drugs for the same illness. They collect independent samples from patients using each drug. Which formula should they use to calculate the standard error for the difference in means, assuming equal population variances?

Accepted Answer

$$ \sqrt{ s_p^2$ \left( \frac{1}{n_1$} + \frac{1}{n_2$} \right) } $$

Question 2

Which of the following best describes the coefficient of determination ($$ R^2$ ) in a $$simple linear regression analysis?

Accepted Answer

The proportion of the total variation in the response variable explained by the regression model.

Question 3

A researcher is testing whether a new teaching method improves test scores. She collects a sample of 40 students and finds a sample mean of 78 with a known population standard deviation of 10. What is the 95% confidence interval for the population mean?

Accepted Answer

$$ 78 \pm 1.96 	imes \frac{10}{\sqrt{40}} $$

Question 4

Which formula would you use to calculate the expected frequency for a cell in a chi-square test of independence?

Accepted Answer

$$ E = \frac{(	ext{row total}) 	imes (	ext{column total})}{	ext{grand total}} $$

Question 5

A hospital administrator wants to estimate the proportion of patients who are readmitted within 30 days. Which formula should be used to construct a 95% confidence interval for the population proportion?

Accepted Answer

$$ \hat{p} \pm z_{\alpha/2} \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} } $$

Question 6

Which of the following is the correct formula for the sample standard deviation?

Accepted Answer

$$ s = \sqrt{ \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$ } $$

Question 7

A study finds a correlation coefficient $$ r = 0.8 $$ between hours studied and exam scores. What is the coefficient of determination, and what does it mean?

Accepted Answer

$$ R^2$ = 0.64 $$, meaning 64% of the variation in exam scores is explained by hours studied.

Question 8

Which of the following is the correct formula for the $$ z -$$score of a value $$ x $$ from a population with mean $$ \mu $$ and standard deviation $$ \sigma $$?

Accepted Answer

$$ z = \frac{x - \mu}{\sigma} $$

Question 9

A biologist is interested in whether the mean weight of a species of fish has changed from the historical mean of 2.5 kg. She collects a sample of 25 fish, finds a sample mean of 2.7 kg, and a sample standard deviation of 0.4 kg. Which test statistic should she use?

Accepted Answer

$$ t = \frac{\bar{x} - \mu_0$}{s/\sqrt{n}} $$

Question 10

Which of the following is the correct formula for the regression line in simple linear regression?

Accepted Answer

$$ \hat{y} = b_0$ + b_1$ x $$

Question 11

A factory quality control manager wants to test if the proportion of defective items is greater than 5%. In a sample of 200 items, 16 are defective. What is the value of the sample proportion $$ \hat{p} $$?

Accepted Answer

$$ \hat{p} = \frac{16}{200} = 0.08 $$

Question 12

Which formula gives the pooled sample variance for two samples with sizes $$ n_1$ $$ and $$ n_2$ $$ and sample variances $$ s_1$^2 $$ and $$ s_2$^2 $$?

Accepted Answer

$$ s_p^2$ = \frac{ (n_1$ - 1)s_1$^2 + (n_2$ - 1)s_2$^2 }{ n_1$ + n_2$ - 2 } $$

Question 13

A clinical trial compares the mean blood pressure reduction between two drugs using independent samples. The sample sizes are large. Which test statistic should be used?

Accepted Answer

$$ z = \frac{ (\bar{x}_1 - \bar{x}_2) - (\mu_1$ - \mu_2$) }{ \sqrt{ \frac{\$$sigma_1$$^2}{n_1$} + \frac{\$$sigma_2$$^2}{n_2$} } } $$

Question 14

Which of the following is the correct formula for the sum of squares due to regression (SSR) in simple linear regression?

Accepted Answer

$$ SSR = \sum (\hat{y}_i - \bar{y})^2$ $$

Question 15

A researcher wants to test if the mean cholesterol level in a population is different from 200 mg/dL. The population standard deviation is unknown, and the sample size is 12. Which distribution should be used to find the critical value?

Accepted Answer

The $$ t -$$distribution with $$ n-1 $$ degrees of freedom.

Question 16

Which formula is used to calculate the standard error of the slope ($$ b_1$ ) in $$simple linear regression?

Accepted Answer

$$ s_{b_1$} = \frac{s}{\sqrt{ \sum (x_i - \bar{x})^2$ } } $$

Question 17

A company wants to test if the variance in delivery times has changed from the historical value of 16 minutes squared. Which test should be used?

Accepted Answer

Chi-square test for a single variance.

Question 18

Which formula gives the probability of exactly $$ k $$ successes in $$ n $$ independent Bernoulli trials with probability $$ p  of $$success?

Accepted Answer

$$ P(X = k) = { n \choose k } p^k (1-p$$)^{n-k}$$ $$

Question 19

A scientist wants to compare the means of three different fertilizers on plant growth. Which statistical test should she use?

Accepted Answer

One-way ANOVA

Comprehensive Study Notes: Descriptive and Inferential Statistics

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