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Exam 3
Introduction to Statistics- Chapters 4.3 and 5
| Question | Answer |
|---|---|
| Given the mean of a normal distribution, how can you find the median? | The mean is equal to the median |
| Describe the inflection points on the graph of a normal distribution. At what x-values are the inflection points located? | The inflection points of a normal distribution are the points where the curve changes from concave down to concave up, or vice versa x-values for the inflection points are mu-sigma and mu+sigma |
| Why is it correct to say "a" normal distribution and "the" standard normal distribution? | "a" is right because there are infinitely many possible normal distributions, each defined by a different mean and standard deviation "the" is right because it refers to a single, specific case with a mean of 0 and a standard deviation of 1 |
| A z-score is 0. Which of the following statements must be true? a. the mean is 0 b. the corresponding x-value is 0 c. the corresponding x-value is equal toe the mean | c. the corresponding x-value is equal toe the mean |
| What requirements must be met in order to approximate a binomial probability distribution with a normal distribution? | np> or equal to 5 and nq>equal to 5 both ensure that expected number of successes and failures are large enough for the approximation to be accurate |
| State the Central Limit Theorem | take sufficiently large random samples from any population, the distribution of the sample means will be approximately a normal distribution, regardless of the population's original distribution |
| Definition of a normal distribution | A continuous probability distribution for a random variable x, the graph of the distribution is called the “normal curve” |
| Properties of a normal distribution | Same mean, median, and mode Bell-shaped Symmetric about the mean Area equals 1 Inflection points: where is changes concavity, located at mu-sigma and mu+sigma Standard normal distribution: mu=0, sigma= 1, x-values of infection points= -1, 1 |
| Definition of geometric distribution | a discrete probability distribution of a random variable x that satisfies the necessary conditions |
| Condition of a geometric distribution | The trial is repeated until the first success Each trial is independent The probability of success is the same for each trial Random variable x represents the trial in which the first success occurs(not the number of success) |
| Equation of a geometric distribution | The probability that the success occurs on trial number x is P(x)= pq^x-1, where q= 1-p |
| How does the normal curve change if the mean or standard deviation changes? | Mean changes(mu): increase: right shift; decrease: left shift Standard deviation changes(sigma):increase: more spread out; decrease: less spread out |
| Sampling distribution of sample mean(how to find the probability of sample mean) | z= (x bar - mu) / ( sigma / square root of n) |
| Definition of sampling distribution | probability distribution of a sample statistic that is formed when random samples of n are repeatedly taken from a population |
| Definition of sampling distribution of sample means | the distribution when the sample statistic is the sample mean |
| Properties of sampling distributions of sample means | The mean of the sample means(mu x bar) is equal to the population mean The standard of deviation of the sample means sigma x bar is equal to the population standard deviation sigma divided by the square root of the sample size n |
| Definition of the Central Limit Theorem | relationship between the sampling distribution of sample means and the population from which the samples are taken |
| Properties of the Central Limit Theorem | Sample of size n where n is greater than or equal to 30 from population with mu and sigma, SDSM approximates a normal distribution Sample is drawn from a normally distributed population then the SDSM is normally distributed |
| Mean of the sample means formula | mu x bar equals mu |
| Variance of the sample means | sigma2 x bar= sigma2/ n |
| Standard deviation of the sample means formula | sigma x bar= sigma/ square root of n |
| Equation to apply the central limit theorem to find the probability of a sample mean | z= value-mean/ standard error= x bar- mu x bar/ (sigma x bar)= x bar- mu x bar/ (sigma/ square root of n) |
| Normal Approximation to Binomial Distribution | If np and nq are greater than or equal to 5, binomial random variable is approximately normally distibuted with mu and sigma |
| How to find the continuity correction | when you use a continuous normal distribution to approximate a binomial probability, you need to move 0.5 unit to the left and right of the midpoint to include all possible x-values in the interval |
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