The Singular Form Of The Word Dice Is Die.

The Singular Form Of The Word Dice Is Die Tom Was Throwing A Six

The singular form of the word “dice” is “die”. Tom was throwing a six-sided die. The first time he threw, he got a three; the second time he threw, he got a three again. What’s the probability of getting a three at the third time? There are 30 table tennis balls in the box: 6 are green, 10 are red, and 14 are yellow. If you shake the box and then randomly select one ball from the box, what’s the probability that you will get a red one? Rachel was flipping a coin with Jerry. She told Jerry: “I am able to get all heads in two tosses.” Jerry laughed at her: “No, the probability of getting two heads at two tosses is only__.” Jennie and Alex both wanted to get a free ticket for a College Music concert. However, the concert staff told them the tickets were limited. Twenty people wanted to attend the concert but only 10 free tickets were left. So the concert center staff decided to use a lottery to decide who would receive the free tickets. What’s the probability of Jennie and Alex both getting free tickets? Laura and Melissa were playing dice. What is the probability of Laura and Melissa both getting a 6? In a statistics class with 36 students, the professor wanted to know the probability that at least two students share the same birthday. The probability will be___ a. 0.1 b. Much smaller than 0.1 c. Much bigger than 0.1 d. Not possible. If you throw a die twice, what is the probability that you will get a one on the first throw or a one on the second throw (or both)? Jerry got a box full of colorful candy balls. There were 50 of them: 20 red, 10 green, 12 yellow, and 8 blue. After shaking the box, he randomly selected 2 candy balls from the box. What is the probability that the first one was blue and the second one was yellow? Jerry got a box full of colorful candy balls. There were 50 of them: 20 red, 10 green, 12 yellow, and 8 blue. After shaking the box, he randomly selected 2 candy balls from the box. What is the probability that the first is blue and the second one is also blue? Which activity could probabilities be computed using a Binomial Distribution? a. Flipping a coin 100 times b. Throwing a die 100 times c. The probability of getting a heart while playing card games d. Grades earned by 100 students on a statistics final exam. Many researchers have argued that the TB skin test is not accurate. Imagine that the TB skin test is only 70% accurate. Sarah is thinking about having the test. Before she has the test, she wonders the probability that she has TB. The probability of Sarah having TB is … a. 70% b. 35% c. 30% d. More information needed. Imagine that the diabetic test accurately indicates the disease in 95% of the people who have it. What’s the miss rate? Which of the following is the probability that subjects do not have the disease but the test result is positive? a. Miss rate b. False positive rate c. Base rate d. Disease rate. In a normal distribution, the median is ____ its mean and mode. a. Approximately equal to b. Bigger than c. Smaller than d. Unrelated to. In a normal distribution, __ percentage of the area under the curve is within one standard deviation of the mean? a. 68% b. 100% c. 95% d. It depends on the values of the mean and standard deviation. A normal distribution with a mean of 15 and standard deviation of 5. 95% of its area is within__ a. One standard deviation of the mean b. Two standard deviations of the mean c. Three standard deviations of the mean d. It depends on the value of the mode. The mean of a standard normal distribution is: a. 0 b. 1.0 c. -1.0 d. 100. The standard deviation of the mean for a standard distribution is: a. 0.0 b. 1.0 c. 100 d. 68%. A normal distribution with a mean of 25 and standard deviation of 5. What is the corresponding Z score for a case having a value of 10? Consider a normal distribution with a mean of 25 and standard deviation of 4. Approximately, what proportion of the area lies between values of 17 and 33? a. 95% b. 68% c. 99% d. 50%. Consider a normal distribution with a mean of 10 and standard deviation of 25. What’s the Z score for the value of 35? For a standard normal distribution, what’s the probability of getting a positive number? a. 50% b. 95% c. 68% d. We cannot tell from the given information. Two-hundred students took a statistics class. Their professor creatively decided to give each of them their Z-score instead of their grade. Rachel got her Z-score of -0.2. She was wondering how well she did on the exam. a. It was very good, much better than almost all of the other students b. It was so-so, but still better than half of the students c. It was not that good, but not at the bottom of the distribution d. It was very bad and she needs to work much harder next time. A researcher collected some data and they form a normal distribution with a mean of zero. What’s the probability of getting a positive number from this distribution? a. 25% b. 50% c. 75% d. We need to calculate the standard deviation and then decide. Which of the following description of distribution is correct? a. A Binomial distribution is a probability distribution for independent events for which there are only two possible outcomes b. You cannot use the normal distribution to approximate the binomial distribution c. Normal distributions cannot differ in their means and their standard deviations. d. Standard normal distributions can differ in their means and in their standard deviations. A toy factory makes 5,000 teddy bears per day. The supervisor randomly selects 10 teddy bears from all 5,000 teddy bears and uses this sample to estimate the mean weight of teddy bears and the sample standard deviation. How many degrees of freedom are there in the estimate of the standard deviation? Imagine you have a population of 100,000 cases. For which of the following degrees of freedom is the closest estimation of the population parameter? a. 4 b. 6 c. 10 d. 1000. Imagine that the average weight of a total of 500 girls in a high school is 35kg. Tom randomly sampled 10 girls and measured their weight. And then he repeated this procedure for three times. The means and standard deviations are listed as following. Which sample estimate shows the least sample variability? a. Sample one: mean=34, SE=5 b. Sample two: mean=30, SE=2 c. Sample three: mean=26, SE=3 d. Sample four: mean= 38, SE=5. For which of the following degrees of freedom is a t distribution closest to a normal distribution? a. 10 b. 20 c. 5 d. 1000. In order to construct a confidence interval for the difference between two means, we are going to assume which of the followings? (Select all that apply) a. The two populations have the same variance. b. The populations are normally distributed. c. Each value is sampled independently from each other value. d. The two populations have similar means. A researcher tries to compare grades earned on the first quiz by boys and girls. He randomly chooses 10 students from boys and 15 students from girls and calculates the confidence interval on the difference between means. How many degrees of freedom will you get in this t distribution? 32. Which of the following choices is not the possible confidence interval on the population values of a Pearson’s correlation? a. (0.3, 0.5) b. (-0.7, 0.9) c. (-1.2, 0.3) d. (0.6, 0.8). Which of the following descriptions of the t distribution is correct? (Select all that apply) a. With smaller sample sizes, the t distribution is leptokurtic b. When the sample size is large (more than 100), the t distribution is very similar to the standard normal distribution c. With larger sample sizes, the t distribution is leptokurtic d. The t distribution will never be close to normal distribution. _______________refers to whether or not an estimator tends to overestimate or underestimate a parameter. ______________ refers to how much the estimate varies from sample to sample. a. Bias; standard error b. Sample variability; Bias c. Mean; standard deviation d. Standard deviation; Mean. Which of the following descriptions of confidence intervals is correct? a. Confidence intervals can only be computed for the mean b. We can only use the normal distribution to compute confidence intervals c. Confidence intervals can be computed for various parameters d. Confidence intervals can only be computed for the population.

Paper For Above instruction

Statistical concepts underpin the understanding of probability, distributions, estimation, hypothesis testing, and various applied statistical methods. This paper discusses essential topics such as the singular form of “dice,” probability calculations involving different scenarios, the mechanics of probability distributions, the normal distribution, and inferential statistics, including confidence intervals and t-distributions, with relevance to research practices.

Understanding Basic Probability and Dice

The question regarding the singular form of "dice" clarifies that "die" is the correct singular term. When considering the probability of rolling a three on a six-sided die, the outcomes are equally likely, with each face having a 1/6 chance. Since previous throws (a three each time) do not influence future throws, this is an example of independent events, and the probability remains constant at 1/6.

Probability of Drawing Colored Balls

When estimating probabilities related to drawing balls from a collection, the calculation of probability hinges on the ratio of favorable outcomes to total outcomes. For instance, if there are 10 green, 20 red, 12 yellow, and 8 blue balls (total 50), the probability of drawing a red ball is 10/50, which simplifies to 1/5 or 20%. Similarly, the probability of selecting a blue first and yellow second involves multiplying probabilities without replacement: the probability that the first ball drawn is blue (8/50), followed by yellow (12/49).

Probability Distributions and Binomial Distribution

Activities like flipping a coin, rolling dice, or drawing cards can be modeled using the binomial distribution, especially when considering the number of successes in fixed trials. For example, flipping a coin 100 times and counting the number of heads fits a binomial model because each flip is independent, with two outcomes (success or failure). In contrast, activities such as dice rolling or card drawing are suited when the probability of success remains constant across trials.

The Accuracy of Medical Tests and Bayes’ Theorem

The probability that a person has a disease given a positive test result depends on the test's sensitivity (true positive rate), specificity (true negative rate), and the prevalence of the disease. Bayesian inference helps in updating the probability of a condition based on test outcomes. For example, if a TB skin test is 70% accurate and the prevalence is low, the probability that an individual actually has TB given a positive result is less straightforward and requires applying Bayes’ theorem.

Normal Distribution: Characteristics and Applications

The normal distribution is characterized by its symmetry around the mean, where median, mean, and mode coincide. Approximately 68% of data falls within one standard deviation of the mean (the empirical rule), and 95% within two standard deviations. Z-scores standardize observations, enabling calculation of proportions and probabilities using standard normal distribution tables. For example, a Z-score of 2 indicates a value two standard deviations above the mean, which corresponds roughly to the 97.5th percentile in a standard normal distribution.

Standard Normal Distribution and Z-Score Calculations

Standard normal distributions have a mean of 0 and a standard deviation of 1. Z-scores translate raw data into standardized scores, facilitating probability calculations. For example, a value of 10 in a distribution with mean 25 and standard deviation 4 corresponds to a Z-score of (10-25)/4 = -3.75, indicating how many standard deviations the value deviates from the mean.

Sample Variability, Confidence Intervals, and Degree of Freedom

Estimating the precision of a statistic involves calculating confidence intervals, which depend on sample size, variability, and distribution shape. The degrees of freedom (df) in t-distributions affect the shape of the curve; with larger df, the t-distribution approaches a normal distribution. For example, when comparing two sample means with small sample sizes, the degrees of freedom are typically calculated as n1 + n2 - 2.

Hypothesis Testing and Correlation

Constructing confidence intervals for correlation coefficients involves intervals that can range from -1 to 1; intervals extending beyond this range are invalid. The interpretation of the correlation depends on the sign and magnitude, with positive values indicating direct relationships. The use of the t-distribution in testing the significance of correlations relies on df derived from sample size.

Bias and Variability in Estimators

Bias refers to whether an estimator systematically over or underestimates a parameter, while standard error measures the variability of an estimator across samples. Accurate estimation of these parameters is fundamental in inferential statistics, affecting the validity of conclusions drawn from data.

Limitations and Assumptions in Statistical Methods

Methods like confidence intervals and t-tests typically assume normality, independence, and homogeneity of variances. Violations of these assumptions can bias results or increase error rates, emphasizing the importance of understanding context and data properties when applying statistical techniques.

Conclusion

Effective statistical analysis involves understanding probability concepts, distribution properties, and the proper application of inferential methods. Recognizing when and how to use different statistical models, distributions, and assumptions ensures accurate interpretation of data and valid research conclusions.

References

• Agresti, A., & Franklin, C. (2017). Statistics: The Art and Science of Learning from Data (4th ed.). Pearson.
• Freedman, D., Pisani, R., & Purves, R. (2007). Statistics (4th ed.). W. W. Norton & Company.
• Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics (9th ed.). W. H. Freeman.
• Ott, L. (2012). Elementary Survey Sampling (6th ed.). Cengage Learning.
• Myers, D. G., Well, A. D., & Lorch, R. F. (2018). Research Design and Statistical Analysis (4th ed.). Routledge.
• Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
• Lehmann, E. L., & Romano, J. P. (2005). Testing Statistical Hypotheses (3rd ed.). Springer.
• Hogg, R. V., & Tanis, E. A. (2015). Probability and Statistical Inference (9th ed.). Pearson.
• Jaccard, J., & Becker, M. A. (2014). Statistics for the Behavioral Sciences. Cengage Learning.
• Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury Press.