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Directions: Use the P-value Approach for all hypothesis tests. Follow all the steps used in hypothesis testing as delineated in the Guided Notes or the textbook and be sure to verify the conditions for each test. Include any StatCrunch, Surfstat, graphing calculator, etc... output used in the process. I am assuming that all questions will be answered using technology-it is not necessary to include by-hand construction of the test statistics. Submitting test statistics only will not earn you full credit for that portion of the problem. Assume that all samples are randomly obtainec statCrunch, Surfstat, graphing aechnology-it is notQuestion 3: The first significant digit in any number is: 1, 2, 3, 4, 5, 6, 7, 8, and 9. Though we may think that each digit would appear with equal frequency, this is not true. Physicist, Frank Benford discovered that in many situations where counts accumulate over time that the first digit in the count follows a particular pattern. The probabilities of occurrence to the first digit in a number, known as Law, are shown below 1st Digit Probability (Benfords Law) 9 301.176.125.097.079.067.058.051.046 Bloombergs web site (www.bloomberg.com) gives information on the stock price and trading volume of the members of the S&P 500 stock index. Suppose a random sample of 100 stocks on a given day were selected from the site, and the first digit of the daily stock volume (total number of shares traded in a given day) was recorded below. 1st Digit Probability (Benfords Law) Stock Volume (frequency) 1 301.176.125.097.079.067.058.051.046 28 16 18 Using a 10% level of significance, test whether the first digits in the stock price follow the distribution of probabilities given by Benfords Law. Be sure to verify the requirements for the test.

math   Statistics-And-Probability   
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We would like to test, whether the age distribution of stock price follows probabilities given (probability distribution) by Benfords Law. The null hypothesis states that a population parameter is equal to a hypothesized value, and the alternative hypothesis states that a population parameter is smaller, greater, or different than the hypothesized value in the null hypothesis Ho: The distributions are the same H1: The distributions are different To find the chi-square statistic of the sample, we need to calculate the expected frequencies for probabilities given by Benfords Law i.e. multiply each probability by total of frequencies of Stock price (which is 100), e.g. for digit 1 it is 0.301 so 0.301 100 30.1 1st DigitProbability Oi 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046 28 30.1 16 17.6 18 12.5 9.7 7.9 6.7 5.8 5.1 4.6 1 100 100 4 6 6 8 6 Total We will have Ei: Expected frequencies and Oi: Observed frequencies The chi square test statistic is given by, (Oi - Ei)2 Subtract each Ei from Oi, square the difference, divide it by Ei, and then find out the total1st DigitProbability Oi Ei(Oi-Ei)A2/Ei 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046 28 30.1 16 17.6 18 12.5 9.7 7.9 6.7 5.8 5.1 4.6 1 100 100 0.146512 0.145455 2.42 0.297938 0.102532 0.073134 0.006897 0.001961 0.426087 3.620515 4 6 6 6 5 6 8 Total So, it is x2 3.6205 Chi-square will havek-1-9-1-8 degrees of freedom, where k is the number of classes The P-value is the probability that a chi-square statistic having 8 degrees of freedom is more extreme than 3.6205. i.e. P( 2 > 3.6205)-1-P( X2 < 3.6205) [Since excel gives left tailed probabilities) We can se CHISQ.DIST(x, deg_freedom, cumulative excel function. Type -CHISQ.DIST (3.6205,8,TRUE). So p-value would be 1- 0.1104 0.8896 Since p-value > 0.10 (10% of level of significance), we accept the null hypothesis Ho: The distributions are the same At 10% level of significance, the evident is sufficient to conclude that the first digits in the stock price follow the distribution of the probabilities given by Benfords Law

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