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2 answers:
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a) 1/6
b) 1/36
c)
1H, 2H, 3H, 4H, 5H, 6H
1T, 2T, 3T, 4T, 5T, 6T
Step-by-step explanation:
a)
The probability of a certain event A to occur is given by
where
a is the number of successfull outcomes, in which event A occurs
n is the total number of possible outcomes
In this problem, the event is
"getting a 6 when throwing a dice once"
We know that the possible outcomes of a dice are six: 1, 2, 3, 4, 5, 6, so we he have
The successfull outcome in this case is only if we get a 6, so only 1 outcome, therefore
So, the probability of this event is
b)
In this case instead, we are throwing the dice twice.
The two throws of the dice are independent events (one does not depend on the other): the probability that two independent events A and B occur at the same time is given by the product of the individual probabilities,
where
is the probability that event A occurs
is the probability that event B occurs
Here we have:
- Event A is "getting a 6 in the first throw of the dice". We already calculated this probability in part a), and it is
- Event B is "getting a 6 in the second throw of the dice". Since the dice has not changed, the probability is still the same, so
Therefore, the probability of getting a 6 on both throws is:
c)
In this problem, we have:
- A dice that is thrown once
- A coin that is also thrown once
The dice has 6 possible outcomes, as we stated in part a):
1, 2, 3, 4, 5, 6
While the coin has two possible outcomes:
H = head
T = tail
So, in order to find all the outcomes of the two events combined, we have to combine all the outcomes of the dice with all the outcomes of the coin.
Doing so, using the following notation:
1H (getting 1 with the dice, and head with the coin)
The possible outcomes are:
1H, 2H, 3H, 4H, 5H, 6H
1T, 2T, 3T, 4T, 5T, 6T
So, we have a total of 12 possible outcomes.
Answer:
i) There is insufficient evidence to support the claim that the standard deviation of the instruments is less than 0.00002 millimeters.
ii)
Step-by-step explanation:
represent the sample standard deviation
represent the significance level for the test
represent the value that we want to test
n=8 represent the sample size
A chi-square test can be used to test "if the standard deviation of a population is equal to a specified value. This test can be either a two-sided test or a one-sided test".
Null and alternatve hypothesis
The system of hypothesis on this case would be given by:
Null Hypothesis:
Alternative Hypothesis:
The statistic to test this is given by:
(1)
Part i) Calculate the statistic
If we replace into formula (1) we got this:
(1)
The critical region for a for a lower one-tailed alternative is given by
The degrees of freedom are given by
And if we use the Chi Square distribution with 7 degrees of freedom we see that
And our critical region would be so on this case we can conclude that we fail to reject the null hypothesis. So it's not enough evidence to conclude that the population standard deviation is less than 0.00002 mm.
Part ii) Calculate the p value
In order to calculate the p value we can do this:
If we compare this value with the significance level (0.01) we see that
This agrees with the conclusion since when the p values is greater than the significance level we FAIL to reject the null hypothesis, same conclusion as part i).