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probability distributions 2024
Our random variable, x, represented discrete values, such as the number of successes. We noted that we could graph a discrete probability distribution as a histogram and that there is a direct correspondence between probability and area. - Correct Answers ✅ ya ex) a discrete probability histogram uses the area of rectangles to represent the probability of each outcome - Correct Answers ✅ , In chapter six, we begin studying continuous probability distributions. Our random variable, x, will represent continuous data such as measurements. The graph of a continuous probability distribution is called a density curve. - Correct Answers ✅ The graph of a continuous probability distribution is called a - Correct Answers ✅ density curve KNOW KNOW: A density curve is the graph of a continuous probability distribution. It must satisfy the following properties: - Correct Answers ✅ 1. The total area under the curve must equal 1.
- Every point on the curve must have a vertical height that is 0 or greater. (That is, the curve cannot fall below the x-axis.)
probability distributions 2024
note; because the total area under the density curve is equal to 1, there is a correspondence between area and probability. note; because the total area under the density curve is equal to 1, there is a correspondence between? - Correct Answers ✅ area and probability there r two specific continuous distributions: Uniform Distributions and Standard Normal Distributions. - Correct Answers ✅ cool uniform or normal Unlike discrete probability distributions, we CANNOT FIND the PX of a single continuous probability distribution value, but instead, we must calculate - Correct Answers ✅ probabilities OVER intervals. A continuous random variable has a uniform distribution if its values are spread evenly over the range of probabilities. The graph of a uniform distribution results in a rectangular shape. with uniform distribution think STRAIGHT LINE or RECTANGLE OF MANY COLUMS AT OR NEAR THE SAME HEIGTH - Correct Answers ✅ a distribution is uniform if every outcome represented by the random variable has the same probability of occurring.
probability distributions 2024
note; ALL of the diff waiting times between 0-5 mins are EQUALLY LIKELY , so it is possible a waiting time could be 1.234567 minutes - Correct Answers ✅ p(x)0.2______________________________ | | | area = 1. | |___0_____1______2_____3______4____ the height of the vertical line is 0.2, the enclosed area is EXACTLY 1. (in general WE SHOULD MAKE THE HEIGHT OF THE VERTICAL LINE IN A UNIFORM DISTRIBUTION EQUAL TO 1/RANGE ie 0.2(5) = 1 for the area of the sqaure Given the uniform distribution illustrated in the figure, find the probability that a randomly selected passenger has a waiting time of at least 2 minutes. p(wait time of at least 2 min) = height x width of the INDICATED AREA in the new shaded figure area = 0.2 x 3 = 0.6 - Correct Answers ✅ 0.2__________________________________ | | area=0.2x3 | | | =0.6 | | | | _____0____1_____2____3____4______
probability distributions 2024
The probability of randomly selecting a passenger with a waiting time of at least 2 minutes is 0.6. the width id 3 because 2-5 is a width of 3 boxes or 3 ticks wide, times 0.2, thats the new area The standard NORMAL distribution is a normal probability distribution with: μ = 0 and σ = 1. MEAN= 0 STD DEV = The total area under its density curve is equal to 1. Notice that the horizontal axis is given in terms of z scores. - Correct Answers ✅ we use statcrunch to solve this The standard NORMAL distribution is a normal probability distribution with: μ =? σ =? - Correct Answers ✅ μ = 0 and σ = 1.
probability distributions 2024
2.CALC
3.NORMAL!!*
- chance the values inside P(X)= window the z score will always be? (2) - Correct Answers ✅ 1. positive
- to the R finding z Scores When Given Probabilities (use statcrunch) so if we know the area to the L of the z score is 0.95 (or 95th percentile) we need to find the Z score? note the area of the z score is 5% or 0.5 bc the other area to the L of it is 95% or 0.95 - Correct Answers ✅ z score = 1. For the standard normal distribution, a CRITICAL VALUE is??? - Correct Answers ✅ a z score separating unlikely values from those that are likely to occur. notation; the expression zα denotes the z score with an area of α to its right. - Correct Answers ✅ ,,
probability distributions 2024
ex) Find the value of z0.025. The notation z0.025 is used to represent the z score with an area of 0.025 to its right. Using statcrunch, we calculate: z0.025 = 1.96. - Correct Answers ✅ with the symmertry of the curve, the z score on the R is z=1.96, with an area of 0.025. On the very left of the curve the z score is -1.96, with an area of 0.025. NOTE THE SYMMETRY conversion formula for not standard normal distribution (ie mean is not 0 std dev is not 1 he idea is that we can take any normal distribution, change every data value to its corresponding z-score, and obtain a standard normal distribution. - Correct Answers ✅ z= x - mean / std dev Procedure for finding probabilities of non-standard normal curves using stat crunch - Correct Answers ✅. Sketch a normal curve, label the mean and any specific x values, then shade the region representing the desired probability.
- Use Stat crunch to determine the probability. Under Stat select Calculator and then select Normal. You need to enter the mean and standard deviation and fill out the window P(x ) =.
probability distributions 2024
? - Correct Answers ✅ A confidence interval is an interval of values used to estimate the true value of a population parameter. We construct confidence intervals around what we call a point estimate, or estimator, of the parameter. We construct confidence intervals around what? - Correct Answers ✅ around what we call a point estimate, or estimator, of the parameter. KNOW: We construct confidence intervals around what we call a point estimate, or estimator, of the parameter. We want our ESTIMATOR TO BE AN UNBIASED estimator of the parameter. - Correct Answers ✅ ex) In the previous slide, instead of thinking of each data value representing the score of a single student, consider each data value to represent the mean of an entire class. So 30 classes took a test, and the data values represent the mean or the variance or some statistic corresponding to each class. This would be an example of a sampling distribution. - Correct Answers ✅ ,, a CONCEPT of a sampling distribution of a statistic is the distribution of?
- Correct Answers ✅ the distribution of all values of that statistic when
probability distributions 2024
all possible samples of the same size are taken from the same population. the actual SAMPLING DISTRIBUTION OF A STATISTIC (ie the sample mean or sample proportion) is: - Correct Answers ✅ is the distribution of all values of the statistic when ALL possible samples of the SAME SIZE N are taken from the same population. The sampling distribution of a statistic is typically represented as a probability distribution. keep in mind: This idea is very abstract, in that we usually do not have access to an entire population. so in general with a sampling distribution-- we start with a popoulation and then we prpceed to draw samples (typically thousands of them of the same size: s1, s2, s3.....s1000. We then calculate some statistic such as the mean for each sample: xbar1, xbar2, xbar3, xbar the SET OF VALUES {XBAR1 (mean of sample 1), XBAR2 (the mean of sample2), XBAR3, XBAR1000.....} is what we call a sampling disstribution, in this case it would be reffered to as THE SAMPLING DISTRIBUTION OF THE MEAN - Correct Answers ✅ ya! For example, let's consider the heights of everyone in a class of 26 students to be an entire population.
probability distributions 2024
note; The distribution of the sample means tends to be a NORMAL distribution. - Correct Answers ✅ ie if we are searching for a sample mean in the question, we can deduce that this will equal the population mean Let's look at a simple ex) of sample mean distribution.. Let's say our population consists of the 3 values: 2, 5, 10. The mean of our population is 17/3 or 5.67. Let's consider samples of size 2. How about {2, 5} and {2, 10}. The mean of 2 and 5 is 3.5 and the mean of 2 and 10 is 6. If we average the two means we get (3.5 + 6)/2 = 4.75; not the population mean of 5.67. But notice, we didn't consider ALL samples of size 2. We left out {5, 10}. The mean of 5 and 10 is 7.5. If we add all 3 means and divide by 3 we get (3.5 + 6 + 7.5)/3 = 5.67, the true population mean. - Correct Answers ✅ Just because we have 10,000 samples in the previous slide, doesn't mean we have all the samples of size 5. For example, think about how many times you would have to roll the die to get 5 one's in a row? We would need 5 of every number, 1 - 6, to get every sample of size 5 possible. - Correct Answers ✅
probability distributions 2024
The sampling distribution of the sample variance is? - Correct Answers ✅ the distribution of all possible sample variances, with all samples having the same sample size n taken from the same population. Sample variances target the value of the population variance. That is, the mean of all the sample variances is the population variance. The expected value of the sample variance is equal to the population variance. SAMPLE VARIANCE = POPULATION VARIANCE - Correct Answers ✅ the mean of all the sample variances = the population variance The distribution of the sample variances tends to be a distribution is SKEWED TO THE R, they are not normal - Correct Answers ✅ The sampling distribution of the proportion is ex) (proportion) 67 out of 100 people prefer dogs to cats, so the proportion of people who prefer dogs to cats is 67/100 =. WHEN ANSWERING PROPORTION Q'S IN STATCRUNCH REMEMBER WE ANSWER IN FRACTIONS - Correct Answers ✅ the distribution of sample proportions, with all samples having the same sample size n taken from the same population.
probability distributions 2024
distributions have means that are equal to the means of their corresponding parameters. These statistics are better in estimating the population parameter. what are the 3 BIASED estimators? ie these DO NOT TARGET THE POPULATION PARAMETER note; the bias with the standard deviation is relatively small in large samples so s is often used to estimate - Correct Answers ✅ 1. Sample medians,
- ranges
- standard deviations are biased estimators. That is they do NOT target the population parameter. However, we are interested in sampling with replacement in our homework for these two reasons: - Correct Answers ✅ 1. When selecting a relatively small sample from a large population, it makes no significant difference whether we sample with replacement or without replacement.
probability distributions 2024
- Sampling with replacement results in independent events that are unaffected by previous outcomes, and independent events are easier to analyze and result in simpler calculations and formulas. The central limit theorem allows us to use a NORMAL distribution for some very meaningful and important applications. - Correct Answers ✅ ,, The Central Limit Theorem tells us that for a population with ANY distribution ---> the distribution of the sample means approaches a NORMAL distribution AS THE SAMPLE SIZE INCREASES. - Correct Answers ✅ CENTRAL LIMIT THOERM for all samples of the same size n, with n > 30, the sampling distribution of x bar can be approximated by a normal distribution with mean and stand dev of sigma / n - Correct Answers ✅ central limit theorm given needs:
- The random variable x has a distribution (which may or may not be normal) with mean μ and standard deviation σ.
probability distributions 2024
the standard deviation of the sample mean )aka the standard ERROR) is? - Correct Answers ✅ ox = o/ sqaure root of n o= sigma KNOW* Practical Rules Commonly Used w/ CLT for when we can approximate normally
- For samples of **SIZE n >30 , the distribution of the sample means can be approximated reasonably well by a NORMAL distribution. The approximation becomes closer to a normal distribution as the sample size n becomes larger.
- If the OG POP is NORMALLY DISTRIBUTED, then for ANY sample size n, the sample means will be normally distributed (not just the values of n larger than 30). - Correct Answers ✅ ex) normal distribution- As we proceed from n = 1 to n = 50, we see that the distribution of sample means is approaching the shape of a normal distribution. - Correct Answers ✅ ,, ex) uniform distribution As we proceed from n = 1 to n = 50, we see that the distribution of sample means is approaching the shape of a normal distribution.
probability distributions 2024
as the sample size gets larger, the variance and std dev decreases - Correct Answers ✅ example u-shaped distribution As we proceed from n = 1 to n = 50, we see that the distribution of sample means is approaching the shape of a normal distribution. variance and std dev are also decreasing here - Correct Answers ✅ note; as the sample size increases, the sampling distribution of sample means approaches a normal distribution. note; as the sample size increases the variation of the sampling distributions decrease. - Correct Answers ✅ as the sample size increases, the sampling distribution of a sample does what?? - Correct Answers ✅ approaches a normal distribution as the sample size increases the variation of the sampling distributions do what? - Correct Answers ✅ decrease
