Multiple Choice:
A die is rolled 600 times. The face with six spots appears 112 times. Is the die biased towards that face, or is this just chance variation? Answer the question in the steps outlined in Problems 1-6.
3) If the null hypothesis were true, the expected number of times the face with six spots appeared would be
a) 112
b) 100
I believe is a. Right?
4) If the null hypothesis were true, the standard error of the number of times the face with six spots appeared would be?
5) What is the P-value (%) of the test? [Please be careful to enter your answer as a percent; that is, if your answer is 50% you should enter 50 in the blank, not 50%, nor 0.5, nor 1/ 2, etc]
3) It's b
Thank you. Do you know the 4 and 5?
4) 9.13
5) 10.38
Thank you very much Fran. Do you know the others i posted?
3) To determine the expected number of times the face with six spots would appear if the null hypothesis were true, we need to calculate the expected value. The expected value is calculated by multiplying the probability of an event occurring by the number of trials. In this case, if the die is fair and unbiased, each face should have an equal probability of 1/6 of appearing on any given roll.
Therefore, to find the expected number of times the face with six spots would appear, we can multiply the probability (1/6) by the total number of rolls (600):
Expected number = Probability * Total number of rolls
Expected number = (1/6) * 600
Calculating this, we get:
Expected number = 100
So, the expected number of times the face with six spots would appear is 100.
Therefore, the correct answer to question 3 is b) 100.
4) To calculate the standard error of the number of times the face with six spots appears, we need to use the formula for standard error:
Standard Error = Square root of (Expected number * (1 - Probability))
In this case, the expected number is 100 and the probability of the face with six spots appearing is 1/6. Plugging in these values into the formula, we get:
Standard Error = Square root of (100 * (1 - 1/6))
Calculating this, we get:
Standard Error ≈ 9.14
Therefore, the standard error of the number of times the face with six spots appears is approximately 9.14.
5) To find the p-value, we need to compare the observed value (112) to the expected value (100) and the standard error.
The p-value is the probability of observing a test statistic as extreme or more extreme than the observed value, assuming the null hypothesis is true. In this case, we can use a one-sample z-test to calculate the p-value.
To calculate the z-score (test statistic), we can use the formula:
z-score = (Observed value - Expected value) / Standard error
Plugging in the values, we get:
z-score = (112 - 100) / 9.14
Calculating this, we get:
z-score ≈ 1.31
Now, we need to find the corresponding p-value from the z-table. Looking up the z-score of 1.31 in the z-table, we find that the corresponding p-value is approximately 0.9032.
Therefore, the p-value is 90.32%.
So, the correct answer to question 5 is 90.32%.