HP340 Midterm Exam Extra Credit Due 3:30PM April Thu 23, 2015. Late assignments will not be accepted (no exceptions).1. The data set ‘distributions.csv’ contains 4 quantitative variables. Use JMP to construct histograms and compute summary statistics (mean, median, SD, IQR) for each of the 4 variables. Comment of the shape of each of the distributions.W: bimodal distributionX: Unimodal, slightly negatively skewed.Y: Positively skewed distribution.Z: Negatively skewed distribution.2. A random sample size of N1 was drawn from a population with =5.2 and the scores of a quantitative variable X collected for each subject in the sample. A Z-based 95% confidence interval for the mean based on this sample is (14.26, 16.34).a. Find N1 and .(attached)b. Compute a 99% Z-based confidence interval for the mean and interpret it.(attached)c. For what values of 0 the null hypothesis H0 = 0 is rejected at the 1% significance level using a two-sided Z-test?(attached) Xd. Assume now that you have a sample of size N2=100 from the same population and that remains the same as in a): for what values of 0 the null hypothesis H0 = 0 would be rejected now? (also at the 1% significance level and using a two-sided Z-test and this new sample?) Explain why there are values of 0 such that H0 = 0 is reject with N=N2 but are not rejected when N= N1?(attached)e. Based on d. what can you say about the p-value for the two-sided z-test of the null hypothesis H0 = 16? Compute exact the p-value.(attached)3. Determine and whether the following statements are True or False and explain why. If false, turn the statement into a true one by changing (adding or removing) as few words aspossible. Explain why the change turns the statement into a true one.a. As the sample size increases the probability of a type I error and the probability of a type II error decreases when testing a null hypothesis at the 5% level. TRUE: As the sample size increases, the closer it becomes to representing a population and thus reducing the type I & II error.b. The central limit theorem states that the sampling distribution of the mean is normal.FALSE: The central limit theorem states that the sampling distribution approaches the normal distribution as sample size increase. Sampling distribution of the mean will approach the normal distribution as sample size increases, but since it is impossible to represent the entire population in one sample, the distribution of the mean will never reach normal distribution.c. If X ~ N(,2) if and only if Z=(X-)/ ~N(0,1) TRUE: X ~ N(μ,σ2) if and only if Z=(X-μ)/σ ~N(0,1) TRUE; The statement is true because this is the general procedure that we get when we convert the problem into an equivalent one dealing with a normal variable measured in standardized deviation units (the standardized normal variable).d. Only a table of the N(0,1) is needed because any probabilities or percentiles relating to other normal distributions can be obtained from it. TRUE: The table of N(0,1) is needed because every set of data has a different set of values. We have to standardize the normal curve, so we make it have a mean of zero and a standard deviation of one. When the curve is standardized, we can use a Z Table to find percentages under the curve.e. The estimated standard error of the mean decreases as the sample size increases.TRUE: this is expected because if the mean at each step is calculated using a lot of data points, then a small deviation in one value will cause less effect on the final mean. f. The sample variance can be made into an unbiased estimate of the population variance by multiplying it by N/(N-1). XTRUE: Since the observed values, on average, fall closer to the sample mean than to the population mean, the standard deviation (calculated using deviations from the sample mean) underestimates the desired standard deviation of the population. Using n-1 instead of n as the divisor corrects for that by making the result a little bit bigger.4. Answer the following questions.a. Explain what the sampling distribution of the mean is and why it is needed to test hypotheses about the mean and construct confidence intervals for the mean. What is the standard error of the mean? What distribution does the sampling distribution of the mean follows approximately?The sampling distribution of the mean is the distribution of values of the sample mean obtained when all possible samples of a size N are drawn from a population and the sample means are calculated for each sample. It is needed to test hypotheses about the mean and construct confidence intervals for the mean because the sampling distribution of the mean helps find the expected population mean among the sample means. The standard error of the mean is the standard deviation of the sampling distribution of the mean. The sampling distribution of the mean approximately follows the normal distribution as the sample size gets larger. This helps find confidence intervals.The standard error of the mean is a measure of sampling error as well as the standard deviation of the sampling distribution of the mean.b. Why is hypothesis testing not an error free technique? In other words why can’t wehave the probability of a type I or type II errors to both be zero?Hypothesis testing is not an error free technique because we test samples, which cannot represent the population from which it was taken as a whole. It cannot be error free since sampling from a population will have a skewed representation. For example, one could sample only extremes and get variables that only represents the extremes of the population.c. What is the difference between unbiasedeness and consistency?Unbiased estimator is a statistic with a sample mean that equals the population mean over an infinite number of random samples, while a consistency estimator is a probability statistic that’s value becomes closer to the population mean as sample size