Question

A statistics professor is at a supermarket waiting in line to buy some groceries. While waiting for the line to move he listened to several people complaining about the long delays. From the different conversions going on he quickly gathers some inform and estimates from the large sample that the mean waiting time is about 12 minutes. He then estimates the population standard deviation to be 1.5 minutes.

1. (a) Explain in detail using full sentences how he should go about finding a 90% confidence interval for the time he has to wait in line to be served. Enumerate each step.

2. Let’s assume that from the question in part (1) it is determined that its margin of error is approximately 3.5 minutes. (a) What does this mean exactly for a 90% confidence interval?

3. Let’s assume now that the professor finished his computation and determined that the 90% confidence interval will be within (8.7 – 15.3) minutes. (a) If he now wants a 95% confidence interval, will the range be bigger or smaller than (8.7 – 15.3) minutes? Explain.

4. It is normal to think that more is better. In the question from part (3) (a) Is the 95% confidence interval better than the 90% interval? Explain. (b) If that is the case why do we not use a 100% confidence interval? Explain.

Answer 1

Solution :

1. First, let's start with a defintion of a 90% confidence interval. A confidence interval helps us to find a range of values for an unknown parameter, in which that parameter can lie 90% of the times.

For this question, in order to calculate the 90% confidence interval, the professor needs to use the following formula:

Here, X(bar) is the average mean of the sample, in this case, 12 minutes. Z is calculated using the value and the z-table. I'll explain how. is also known as the probability of Type-1 error in statistics. It is usually given in the question. In this question, we need to calculate the 90% confidence interval. Therefore, = 1-0.9 = 0.1. Now, google a z-table and open it. It will look something like this

Now, as you can see, I have highlighted some terms. In the table, the closest value to 0.1 has been highlighted. And the column and row value also have been highlighted. Thus, Z  = 1.28. Also, = 1.5 minutes. However, in this question, the value of n is not given. We need to have n to calculate the value of 90% confidence interval.

Answer 2. In this question, we have the margin of error as 3.5 minutes. The formula for margin of error is:

As you can see, this forms the later part of the formula of the confidence interval. The confidence interval is given as: [12-3.5, 12+ 3.5] = [8.5, 15.5].

Answer 3. If we take a 95% confidence interval, = 1 - 0.95 = 0.05. From the z-table above, you can calculate Z = 1.65. Even if we don't do the calculations, we can see the interval would be narrower. The reason behind this is that whenever, we calculate a 95% confidence interval, we calculate the interval on a better level of precision, which technically means that the interval would be more precise, hence narrower as compared to a 90% interval.

Answer d. I have already explained the first part of the question. For the second part, consider the logic of a normal distribution. In the normal distribution. We know that it covers values from −∞→∞. This means no matter how wide your range is constructed you will never include all the possible numbers. Simply put you can never bee 100% certain you captured the true population value because −∞≤n≤∞ and any bounded interval will clearly not cover all the possible values.

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