is the lottery fair? powerball is a multistate lottery in which players try to guess the numbers that will…

is the lottery fair? powerball is a multistate lottery in which players try to guess the numbers that will turn up in a drawing of numbered balls. one of the balls drawn is the \powerball.\ matching the number drawn on the powerball increases ones winnings. in a 17 - month period, the powerball was drawn from a collection of 35 balls numbered 1 through 35. a total of 147 drawings were made. for the purpose of this exercise, we grouped the numbers into five categories: 1 - 7, 8 - 14, and so on. if the lottery is fair, then the winning number is equally likely to occur in any category. following are the observed frequencies. can you conclude that the categories are not equally likely? use the 0.010 level of significance and the p - value method with the ti - 84 plus calculator.\ncategory|1 - 7|8 - 14|15 - 21|22 - 28|29 - 35\nobserved|25|30|34|23|35\nsend data to excel\npart 1 of 7\n(a) state the null and alternate hypotheses.\n$h_0:p_1 = p_2 = p_3 = p_4 = p_5=0.2$\n$h_1:$ some or all of the actual probabilities differ from those specified by $h_0$.\npart: 1 / 7\npart 2 of 7\n(b) compute the expected frequencies.\ncategory|1 - 7|8 - 14|15 - 21|22 - 28|29 - 35\nexpected| | | | |

is the lottery fair? powerball is a multistate lottery in which players try to guess the numbers that will turn up in a drawing of numbered balls. one of the balls drawn is the \powerball.\ matching the number drawn on the powerball increases ones winnings. in a 17 - month period, the powerball was drawn from a collection of 35 balls numbered 1 through 35. a total of 147 drawings were made. for the purpose of this exercise, we grouped the numbers into five categories: 1 - 7, 8 - 14, and so on. if the lottery is fair, then the winning number is equally likely to occur in any category. following are the observed frequencies. can you conclude that the categories are not equally likely? use the 0.010 level of significance and the p - value method with the ti - 84 plus calculator.\ncategory|1 - 7|8 - 14|15 - 21|22 - 28|29 - 35\nobserved|25|30|34|23|35\nsend data to excel\npart 1 of 7\n(a) state the null and alternate hypotheses.\n$h_0:p_1 = p_2 = p_3 = p_4 = p_5=0.2$\n$h_1:$ some or all of the actual probabilities differ from those specified by $h_0$.\npart: 1 / 7\npart 2 of 7\n(b) compute the expected frequencies.\ncategory|1 - 7|8 - 14|15 - 21|22 - 28|29 - 35\nexpected| | | | |

Answer

Explanation:

Step1: Recall expected - frequency formula

The formula for the expected frequency $E_i$ in a goodness - of - fit test is $E_i = n\times p_i$, where $n$ is the total number of observations and $p_i$ is the probability of the $i$th category. Here, $n = 147$ (total number of drawings) and $p_i=0.2$ for each of the 5 categories.

Step2: Calculate expected frequencies for each category

For the category 1 - 7: $E_1=n\times p_1 = 147\times0.2 = 29.4$ For the category 8 - 14: $E_2=n\times p_2 = 147\times0.2 = 29.4$ For the category 15 - 21: $E_3=n\times p_3 = 147\times0.2 = 29.4$ For the category 22 - 28: $E_4=n\times p_4 = 147\times0.2 = 29.4$ For the category 29 - 35: $E_5=n\times p_5 = 147\times0.2 = 29.4$

Answer:

Category 1 - 7 8 - 14 15 - 21 22 - 28 29 - 35
Expected 29.4 29.4 29.4 29.4 29.4