Probability Distributions
What is the sum of the probabilities in a probability distribution?
1
What is the shape of a normal distribution curve?
Bell-shaped
What term describes the distribution of a statistic, such as the sample mean or sample proportion, calculated from multiple samples of the same size taken from the same population?
Sampling Distribution
What is a confidence interval?
A range of values used to estimate the true value of a population parameter.
What is the null hypothesis in hypothesis testing?
The hypothesis that there is no effect or difference.
What type of probability distribution is used to model the number of successes in a fixed number of independent Bernoulli trials?
Binomial Distribution
What is the empirical rule in a normal distribution?
Approximately 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.
What theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases?
Central Limit Theorem
What does a 95% confidence interval mean?
There is a 95% probability that the interval contains the true population parameter.
What is the alternative hypothesis in hypothesis testing?
The hypothesis is that there is an effect or a difference.
What is the name of the distribution that has parameters n (number of trials) and p (probability of success)?
Binomial Distribution
What does the standard normal distribution's mean equal?
0
For a population with a known mean μ and standard deviation σ, what is the standard deviation of the sampling distribution of the sample mean (also known as the standard error)?
σ / (√n)
What is the margin of error in a confidence interval?
The amount added and subtracted from the sample statistic to create the interval.
E = Z * (σ / √n)
What is a Type I error in hypothesis testing?
Rejecting the null hypothesis when it is actually true.
In a geometric distribution, what does the variable X represent?
The number of trials until the first success
What is the z-score formula?
z=(X−μ) / σ
When the sample size is large, which distribution can the distribution of the sample proportion be approximated by?
Normal Distribution
What is the formula for a confidence interval for a population mean when the population standard deviation is known?
X ± Z * (σ / √n)
A researcher conducts a hypothesis test with a significance level (alpha) set at 0.05. After analyzing the data, the calculated p-value is 0.03. What can the researcher conclude from this result?
The researcher can conclude that since the p-value (0.03) is less than the significance level (0.05), there is sufficient evidence to reject the null hypothesis at the 0.05 level of significance.
What is the formula for the mean of a binomial distribution?
μ=np
A normally distributed dataset has a mean of 50 and a standard deviation of 10. What is the probability that a randomly selected value from this dataset is greater than 65?
0.0668 or 6.68%.
A factory produces light bulbs with a mean lifespan of 1000 hours and a standard deviation of 50 hours. A random sample of 50 light bulbs is taken from the production line. Determine whether the Central Limit Theorem applies to the distribution of the sample mean lifespan.
Yes, since the sample size is n = 50, which is greater than30, the Central Limit Theorem applies.
How does increasing the sample size affect the width of the confidence interval?
It decreases the width of the confidence interval.
A researcher wants to determine if a new study method improves students' performance on a standardized test. The null hypothesis (H0) states that the new method has no effect, while the alternative hypothesis (H1) claims that it does improve performance. The researcher selects a random sample of 25 students and records their test scores before and after implementing the new method. The average difference in scores is 3 points, and the standard deviation of the differences is 1.2 points. What is the researcher's conclusion?
The t-value is 12.5. The P-value is near 0. Because the p-value is smaller than the significance level 0.001, the researcher can reject the null hypothesis. The researcher can conclude that the new study method significantly improves students' performance on standardized tests.