Unit 5 MATHS & STATISTICS ALL SOLUTION LATEST 2023/24 EDITION GUARANTEED GRADE A+

Sample Statistic Measure of an attribute of a sample. Sample proportion = P(hat) Sample Mean = X (Line over it) Sample Std Dev = s sample mean is the sum of a certain attribute of a sample divided by the sample size. Sample Mean A mean obtained from a sample of a given size. Denoted as X(line over it) Population Parameter Corresponding measurement for the population. Something that we can find in a sample. The only way to figure out a parameter is to take a census. Population Mean Denoted as the greek letter mu Population proportion = p Population Mean=. greek U Population Std Dev = fancy o Parameters a numerical value that characterizes some aspect of a population. Example: Average number of all adults who smoke regularly Symbols: u , o Statistics sample measures that we can use to estimate parameters, which are corresponding population measure. It's important to remember that this only works when the sample is carried out well. For instance, if there's bias, then those wouldn't accurately reflect the population measures. Sampling with replacement means that you put everything back once you've selected it. Typically, one big requirement for statistical inference is that the individuals, the values from the sample, are independent. One doesn't affect any of the others. When sampling with replacement, each trial is independent. sampling without replacement which means that each observation is not put back once it's selected--once it's selected, it's out and cannot be selected again. For independence, a large population is going to be at least 10 times larger than the sample. Sampling Error simply relates to the variability within the sampling distribution. It is the amount by which the sample statistic differs from the population parameter. Standard deviation decrease as the sample size increases. Select the False statement about sampling error and sample size The standard error increases as the sample size increase. True: The sample size can impact the sampling error The larger a sample size, the more accurate an estimate can be In order to decrease sampling error, you must increase the sampling size Distribution of Sample Means Distribution that shows the means from all possible samplings of a given size. Each data points consist of a mean of a collected sample. Every possible sample mean will be plotted in the distribution. Represent all the distrubtions Step 1: Take the sample means and graph them. Draw out an axis. Step 2: Take the average value, for example, the mean of 2.5, and put a dot at 2.5 on the x-axis much like a dot plot. Do this for all the sample means that you have found. Step 3: Keep doing this over and over again. Ideally you would do this hundreds or thousands of times to show the distribution of all possible samples that could be taken. Standard Deviation of Sampling Distribution of Sample Means Also called the standard error. IT's the standard deviation of the population, divided by the square root of the sample size. Standard Error Standard deviation of the sampling distribution of sample means Select the true statement between sample size and the standard deviation of distribution of sample means. D a.)Sample size does not have an impact on standard error. b.)As sample size increases, standard error increases. c.)As sample size decreases, standard error decreases. d.)As sample size increases, standard error decreases. Proportions The only way to obtain the true population proportion, which is the parameter we're trying to estimate, is by taking a cenus. Distribution of Sample Proportions Distribution of all possible sample proportions for a certain size, n. A distribution where each data point consists of a proportion of successes of a collected sample. For a given sample size, every possible sample proportion will be plotted in the distribution. Mean of Distribution of Sample Proportions Value of P, which is the actual probability. Proportion of Successes Number of successes divided by the sample size, n. The Standard deviation will be the standard deviation of the binomial distribution, divided by n. Standard Deviation of a Distribution of Sample Proportions Square root of n times p times q, divded by n. Also known as the standard error. A measure calculated by taking the square root of the quotient of p(1-p) and n. Hint If p is the probability of success, q is the probability of failure, which is equal to 1-p. Jeanette really loves apple-flavored Fruity Tooty candies, but there always seems to be a lot of cherry-flavored candies in each bag. To determine whether this is because cherry candies are so popular or because each bag contains fewer apple candies, Jeanette randomly picks a Fruity Tooty candy from her bag, records its flavor, and places it back in the bag. Each bag contains a mixture of cherry, grape, apple, lemon, and orange flavors. Which statement about Jeanette's distribution of sample proportions is true? D a.)The distribution of the count of picking a cherry candy cannot be modeled as approximately normal if Jeanette picks candies over 100 times. b.) The distribution of the count of picking a lemon candy can be modeled as approximately normal if Jeanette picks candies 15 times. c.) The count of drawing an apple candy is not a binomial distribution. d.)The count of drawing an orange candy has a binomial distribution. Recall that proportions can be put into two categories. If we assume that items are independently chosen, this would follow the binomial distribution. Theresa really loves orange-flavored Fruity Tooty candies, but there always seems to be a lot of cherry-flavored candies in each bag. To determine whether this is because cherry candies are so popular or because each bag contains fewer orange candies, Theresa randomly picks a Fruity Tooty candy from her bag, records its flavor, and places it back in the bag. Each bag contains a mixture of cherry, grape, apple, lemon, and orange flavors.Which statement about Theresa's distribution of sample proportions is true? C. a.)The count of drawing an orange candy is not a binomial distribution. (Not this because this would have two categories, orange and not orange.) b.) The distribution of lemon candy can be modeled as normal if Theresa picks candies 12 times. c.) The sample proportion of cherry candies has a binomial distribution. d.) The distribution of the count of picking an apple candy cannot be modeled as approximately normal if Theresa picks candies over 300 times. Recall proportions can be put into two categories. IF we assume that items are independently chosen, this would follow the binomial distribution. She can either pick a cherry candy or not pick a cherry candy. Susan really loves the lemon-flavored Fruity Tooty candies, but there always seems to be a lot of grape-flavored candies in each bag. To determine whether this is because grape candies are so popular or because each bag contains fewer lemon candies, Susan randomly picks a Fruity Tooty candy from her bag, records its flavor, and places it back in the bag. Each bag contains a mixture of cherry, grape, apple, lemon, and orange flavors.Which statement about Susan's distribution of sample proportions is true? D. a.) The distribution of the count of picking a cherry candy cannot be modeled as approximately normal if Susan picks candies over 200 times. b.)The sample proportion of drawing an orange candy is not a binomial distribution. c.) The distribution of lemon candy can be modeled as normal if Susan picks candies 10 times. d.) The distribution of apple candy can be modeled as normal if Susan picks candies over 75 times. In General the sampling distribution of the proportions follows a binomial distribution. As you let the sample size get larger, we can use a normal approximation to the binomial and note that the sampling distribution converges to normal distribution. Hypothesis tsting Standard procedure in statistics for testing a hypothesis, or claim about population parameters. Null Hypothesis A claim about a particular value of a population parameter that serves as the starting assumption for a hypothesis test. Ho Alternative Hypothesis A claim that a population parameter differs from the value claimed in the null hypothesis. Ha A school authority claims that the average percentage marks of students is 68. A researcher has taken a well-designed survey and his sample mean is 64.5 and sample standard deviation is 2. The sample size is 300.Which statement is correct? B a.) The difference exists due to chance since the test statistic is small. b.) The result of the survey is statistically significant. c.) The sample size should be much more. d.) The result of the survey is not statistically significant. Since we find that 64.5 is much lower than the average 68 from a large sample size and relatively small standard deviation, we can conclude there is a statistically significant result. It is also practically different as well. A school authority claims that the average height of students is 178 cm. A researcher has taken a well-designed survey and his sample mean is 177.5 cm and the sample standard deviation is 2. The sample size is 25.Which statement is correct? A a.) The difference exists due to chance since the test statistic is small. b.) The sample mean and population mean is the same. c.) The result of the survey is statistically significant. d.) The result of the survey is biased. With a very small sample size of 25, a difference of 0.5 cm is most likely due to chance. Type 1 Error Error that occurs when a true null hypothesis is rejected. Type II Error An error that occurs when a false null hypothesis is not rejected. A spice box manufacturing company is having difficulty filling packets to the required 50 grams. Suppose a business researcher randomly selects 60 boxes, weighs each of them and computes its mean. By chance, the researcher selects packets that have been filled adequately and that is how he gets the mean weight of 50 g, which falls in the "fail to reject" region.The decision is to fail to reject the null hypothesis even though population mean is NOT actually 50 g.Which kind of error has the researcher done in this case? D a.)Neither b.)Type I c.)Both d.)Type II Since the sample evidence provides evidence the null should not be rejected, when in fact the null hypothesis is false, this is called Type II error. Significance Level The probability of making a Type 1 Error. Abbreviated with the symbol alpha, a. Being comfortable making some error sometimes. Power of a Hypothesis Test Probability of rejecting the null hypothesis correctly, rejecting when the null hypothesis is false, which is a correct decision. Probability that we reject the null hypothesis when a difference truly does exist. Which of the following statements is FALSE? Between Significance level and the power of a hypothesis test? A. a.) Reducing the significance level (α) can increase a test's effectiveness. b.) Alpha (α) is equal to the probability of making a Type I error. c.) ​Expanding the sample size can increase the power of a hypothesis test. d.) A larger sample size would increase the effectiveness of a hypothesis test Recall that alpha is Type I error and corresponds to the threshold we use of acceptable Type I error. So if we increase alpha we are opening ourselves up to more Type I error. This is not an increase in the test's effectiveness. Which of the following statements is FALSE? Between significance level and the power of a hypothesis test. A. a.) Expanding the sample size can decrease the power of a hypothesis test. b.) The probability of rejecting the null hypothesis in error is called a Type I Error. c.) A larger sample size would increase the power of a significance test. d.) The significance level is the probability of making a Type I error. If you increase the sample size, the ability to detect differences increases, which reduces Type II error. So this means the power of the test has increased. One tailed Test A test for when you have a reason to believe the population parameter is higher or lower than the assumed parameter value of the null hypothesis. Right Tailed Left Tailed Only test whether or not there is evidence of a statistic being significantly higher or lower than a claimed parameter., like mu or p. Right Tailed Test Type of one tailed test that means that the alternative hypothesis is larger than the claimed parameter. The distribution of a right tailed test would look similar to the following curve. Left Tailed Test Type of One tailed test in which alternative hypothesis claims that it's less than the claimed parameter. Looking at the values lower than the assumed value, which is the section to the left of the value. Two Tailed Test When we have no reason to believe the population is different from the assumed parameter value of the null hypothesis. Looking at the values on the values that are extremely lower or higher than the assumed value. Not equal to. Test Statistic Relative distance of the statistic obtained from the sample from hypothesized value of the parameter from the null hypothesis. Measured in terms of the number of standard deviations from the mean of the sampling distribution. Z-Statistic or Z-Score Measurement in standardized units of how far a sample statistic is from the assumed parameter if the null hypothesis is true. Test Statistic Formula Statistic - parameter standard deviation of statistic Z-Statistic of Means Statistic : Sample Mean X_, Parameter: Hypothesized population mean: u, Standard Deviation of x_: o divided by square root of n. Formula: x-u divided by o that is divided by the square root of n. Z-Statistic for Proportions Statistic: Sample proportion: phat Parameter: Hypothesized population proportion: p Standard Deviation: Phat: quare root of pq divided by n. The standard deviation of the phat statistic is going to be the square root of p times q which is 1-p, over n. Therefore, the z-statistic for sample proportions that you can calculate is your test statistic, and it is equal to phat minus p from the null hypothesis, divided by the standard deviation of phat. Select the correct statement. Determine whether to reject a null hypothesis from a given p value and significance level. D a.)Given a p-value of 0.05, and a significance level of 3%, you should reject the null hypothesis.​ b.)Given a p-value of 0.08, and a significance level of 2%, you should reject the null hypothesis.​ c.)Given a p-value of 0.06, and a significance level of 5%, you should reject the null hypothesis.​ d.)Given a p-value of 0.01, and a significance level of 5%, you should reject the null hypothesis.​ CONTINUED..