Q: In a medical study, the probability of a patient having disease A is $$P(A)$$ and the probability of having disease B is $$P(B). If $$the diseases are independent, what is the probability that a patient has both diseases?

$$P(A \cap B) = P(A) \times P(B)$$

Question 1

A pharmaceutical company wants to estimate the average effect of a new drug on blood pressure. They collect a sample of $$n$$ patients and record their blood pressure reduction values $$x_1$$, $$x_2$$, ..., x_n$$. Which formula should they use to calculate the sample mean reduction?

Accepted Answer

$$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$$

Question 2

A researcher wants to measure the variability in exam scores for a class of 30 students. Which formula should be used to calculate the sample variance?

Accepted Answer

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

Question 3

Which formula represents the sample correlation coefficient $$r_{xy}$$ between two variables $$x$$ and $$y$$?

Accepted Answer

$$r_{xy} = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^n (x_i - \bar{x})^2$ \sum_{i=1}^n (y_i - \bar{y})^2$}}$$

Question 4

A scientist is analyzing the relationship between temperature and reaction rate using regression. What is the formula for the regression equation?

Accepted Answer

$$y = a + bx$$

Question 5

In a medical study, the probability of a patient having disease A is $$P(A)$$ and the probability of having disease B is $$P(B). If $$the diseases are independent, what is the probability that a patient has both diseases?

Accepted Answer

$$P(A \cap B) = P(A) 	imes P(B)$$

Question 6

A factory produces light bulbs, and the probability that a bulb is defective is $$0.02. If 100 $$bulbs are tested, which distribution best models the number of defective bulbs?

Accepted Answer

Binomial distribution

Question 7

A hospital administrator wants to estimate the mean waiting time for patients. The sample mean is $$\bar{x}$$ and the sample standard deviation is $$s. $$What is the formula for the standard error of the mean?

Accepted Answer

$$\frac{s}{\sqrt{n}}$$

Question 8

A researcher wants to test if a new teaching method improves test scores. The null hypothesis is $$H_0$$: \mu = \$$mu_0. $$Which formula is used to calculate the test statistic for a one-sample $$z-$$test?

Accepted Answer

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

Question 9

A clinical trial compares the mean recovery times of two treatments. The sample means are $$\bar{x}_1$$ and $$\bar{x}_2$$, and the sample variances are $$s_1^2$ and s_2$^2$$ with sample sizes $$n_1$$ and $$n_2. $$What is the formula for the standard error of the difference in means for independent samples?

Accepted Answer

$$\sqrt{\frac{s_1$^2}{n_1$} + \frac{s_2$^2}{n_2$}}$$

Question 10

A biologist wants to construct a 95% confidence interval for the mean height of a plant species. Which formula should be used?

Accepted Answer

$$\bar{x} \pm z_{\alpha/2} \frac{s}{\sqrt{n}}$$

Question 11

A psychologist is interested in the probability that a randomly selected person scores between $$a$$ and $$b on a $$standardized test, assuming scores are normally distributed with mean $$\mu$$ and standard deviation $$\sigma. $$Which formula should be used?

Accepted Answer

$$P(a \u003c X \u003c b) = \int_a^b \frac{1}{\sigma \sqrt{2\pi}} e$$^{-\frac{(x-\mu)^2}$${2\$$sigma^2$$}} dx$$

Question 12

A company wants to test if the proportion of defective products has changed from the historical rate $$p_0. $$Which formula is used for the test statistic in a one-sample proportion $$z-$$test?

Accepted Answer

$$z = \frac{\hat{p} - p_0$}{\sqrt{\frac{p_0$(1-p_0$)}{n}}}$$

Question 13

A nutritionist compares the mean calorie intake between two groups using independent samples. What is the formula for the confidence interval for the difference in means?

Accepted Answer

$$(\bar{x}_1 - \bar{x}_2) \pm z_{\alpha/2} \sqrt{\frac{s_1$^2}{n_1$} + \frac{s_2$^2}{n_2$}}$$

Question 14

A scientist wants to test if two drugs have different effects on recovery time using paired samples. What is the formula for the test statistic for the mean difference?

Accepted Answer

$$t = \frac{\bar{d}}{s_d / \sqrt{n}}$$

Question 15

A researcher is testing the null hypothesis $$H_0$$: \$$mu_1$$ = \$$mu_2$$ for two independent samples. Which formula is used for the test statistic if the population variances are assumed equal?

Accepted Answer

$$t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1$} + \frac{1}{n_2$}}}$$

Question 16

A public health official wants to compare the proportion of smokers in two cities. What is the formula for the standard error of the difference in proportions?

Accepted Answer

$$\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1$} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2$}}$$

Question 17

A scientist is conducting a hypothesis test with significance level $$\alpha. $$What is the critical value for a two-tailed $$z-$$test?

Accepted Answer

$$z_{\alpha/2}$$

Question 18

A researcher wants to calculate the pooled variance $$s_p^2$ for two samples with sizes n_1$ and n_2$ and sample variances s_1$^2$$ and $$s_2^2$. Which formula should be used?

Accepted Answer

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

Statistics I: Comprehensive Formula Sheet and Study Guide

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