A six-sided die is rolled 20 times and a 5 is rolled 9 times.
a) State a null and alternative hypothesis for the die being fair.
b) Can the null hypothesis be rejected at the .01 significance level?
a fair die is toss 30 times what is the probabilities
an eight-sided die,has eight faces and number 1,2,3,4,5,6,7,8,respectively. assume that this die is fair and x equals the outcome when it is rolled ones.a) describe the space of x .b) find p(x<3),p(x>5),p(2<x<4)
a) The null hypothesis (H0) for the die being fair would be that the probability of rolling a 5 is equal to the expected probability of rolling any other number on the die (1/6 = 0.1667). The alternative hypothesis (H1) would be that the probability of rolling a 5 is not equal to 0.1667.
b) To determine if the null hypothesis can be rejected at the 0.01 significance level, we need to perform a hypothesis test using the binomial distribution.
First, we calculate the expected number of 5s rolled if the die were fair. Since there are 20 rolls and each roll has a 1/6 probability of landing on a 5, the expected number of 5s rolled would be:
Expected number of 5s = (Total number of rolls) * (Probability of rolling a 5) = 20 * (1/6) = 3.3333
Next, we need to calculate the probability of observing 9 or more 5s in 20 rolls, assuming the null hypothesis is true (i.e., the die is fair). We use the binomial distribution formula with n = 20 (number of trials), p = 1/6 (probability of success - rolling a 5), and x ≥ 9 (number of successful trials):
P(X ≥ 9) = 1 - P(X < 9) = 1 - P(X ≤ 8)
To find P(X ≤ 8), we can use software or tables to calculate the cumulative probability.
Finally, compare the calculated probability with the significance level. If the calculated probability is less than the significance level, the null hypothesis can be rejected. If the calculated probability is greater than or equal to the significance level, we fail to reject the null hypothesis.
Performing these calculations, we can determine if the null hypothesis can be rejected at the 0.01 significance level.