By iteratively substituting, we have



and the pattern continues down to the first term
,



Recall the formulas


It follows that



<h3>
Answer: Independent</h3>
For two events A and B, if the occurrence of either event in no way affects the probability of the occurrence of the other event, then the two events are considered to be <u> independent </u> events.
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Explanation:
Consider the idea of flipping a coin and rolling a dice. If these actions are separate (i.e. they don't bump into each other), then one object won't affect the other. Hence, one probability won't change the other. We consider these events to be independent.
In contrast, let's say we're pulling out cards from a deck. If we don't put the first card back, then the future probabilities of other cards will change. This is considered dependent.
Answer:

And replacing we got:

And we can calculate this probabilit using the normal standard distribution or excel and we got:

Step-by-step explanation:
If we define the random variable of interest "the amount spent by a family of four of food per month" and we know the following parameter:

And we want to find the following probability:

And we can use the z score formula given by:

And replacing we got:

And we can calculate this probabilit using the normal standard distribution or excel and we got:

Answer:
a. With 90% confidence the proportion of all Americans who favor the new Green initiative is between 0.6290 and 0.6948.
b. If the sample size is changed, the confidence interval changes as the standard error depends on sample size.
About 90% percent of these confidence intervals will contain the true population proportion of Americans who favor the Green initiative and about 10% percent will not contain the true population proportion.
Step-by-step explanation:
We have to calculate a 90% confidence interval for the proportion.
The sample proportion is p=0.6619.

The standard error of the proportion is:

The critical z-value for a 90% confidence interval is z=1.6449.
The margin of error (MOE) can be calculated as:

Then, the lower and upper bounds of the confidence interval are:
The 90% confidence interval for the population proportion is (0.6290, 0.6948).