Gasoline prices reached record high levels in 16 states...

Gasoline prices reached record high levels in 16 states during 2003 (The Wall Street Journal, March 7,2003 ). Two of the affected states were California and Florida. The American Automobile Association reported a sample mean price of $\$ 2.04$ per gallon in California and a sample mean price of $\$ 1.72$ per gallon in Florida. Use a sample size of 40 for the California data and a sample size of 35 for the Florida data. Assume that prior studies indicate a population standard deviation of .10 in California and .08 in Florida are reasonable. a. What is a point estimate of the difference between the population mean prices per gallon in California and Florida? b. At $95 \%$ confidence, what is the margin of error? c. What is the $95 \%$ confidence interval estimate of the difference between the population mean prices per gallon in the two states?

Gasoline prices reached record high levels in 16 states during 2003 (The Wall Street Journal, March 7,2003 ). Two of the affected states were California and Florida. The American Automobile Association reported a sample mean price of $\$ 2.04$ per gallon in California and a sample mean price of $\$ 1.72$ per gallon in Florida. Use a sample size of 40 for the California data and a sample size of 35 for the Florida data. Assume that prior studies indicate a population standard deviation of .10 in California and .08 in Florida are reasonable.
a. What is a point estimate of the difference between the population mean prices per gallon in California and Florida?
b. At $95 \%$ confidence, what is the margin of error?
c. What is the $95 \%$ confidence interval estimate of the difference between the population mean prices per gallon in the two states?

Key Concepts

Point Estimate

The point estimate of the difference between two population means is obtained by subtracting one sample mean from the other. This gives a single best estimate of the true difference between the two population parameters based on the data available.

Standard Error of the Difference

When population standard deviations are assumed known, the standard error for the difference in means is calculated by taking the square root of the sum of each population variance divided by its corresponding sample size. This quantifies the variability in the difference estimate due to sampling.

Margin of Error

The margin of error is determined by multiplying the critical value (from the normal distribution when population parameters are known) by the standard error. It represents the extent to which the point estimate might differ from the true population difference with a specified level of confidence.

Confidence Interval

A confidence interval for the difference between population means is a range constructed by adding and subtracting the margin of error from the point estimate. It provides a likely range for the true difference between the population parameters with a particular level of confidence (e.g., 95%).

Transcript

00:01 Once again, welcome to a new problem.

00:04 This time you're dealing with confidence intervals.

00:08 And whenever you make predictions in statistics, you are dealing with confidence intervals.

00:18 So what you're saying is that you could either have estimation, you could either have estimation or hypothesis testing.

00:30 Hypothesis testing, you could have null and alternative hypothesis.

00:34 When it comes to estimation, you could use confidence interval where you're either say 90 % confident that something is going to happen or you're going to make a certain prediction or you're 95 % confident or your 99 % confidence.

00:51 So these are the things that are going to happen when it to confidence intervals and your goal in making predictions is for example to see that the mean of a sample which you call x bar this is the mean of a sample predicts the mean of the population with a certain level of confidence of course your errors allowed to happen so your x bar will have have some kind of margin of error when you're running a confidence interval.

01:33 And so the purpose of building an interval is to say if your mean is a certain value, you're going to have some intervals that capture the mean and then you're also going to have some intervals that don't capture the mean.

01:50 But if you're 95 % confident, which is the default interval, you'll find that of all these intervals, 95 out of 100 of them are going to capture the mean.

02:03 Intervals are not just for making these kinds of predictions.

02:09 You could also have a problem like the case we have where we're dealing with gas prices.

02:15 Gas prices, say in 2003, we have gas prices and you're doing a comparative analysis between california and florida when it comes to gas prices.

02:30 And in california, your x bar for gas prices, which is the mean of a sample, is 2 .04.

02:39 And then for the case of florida, the gas is a little bit cheaper.

02:44 The overall average price of gases when you take a sample is $1 .72.

02:53 So it's $1 .72.

02:55 Remember, you're using the same sample size in this case.

03:01 And don't forget that when it comes to the data, the sample size for california is a little bit bigger than the sample size of florida, which is 35.

03:17 And then of course, we do have population standard deviation for california is 10 cents.

03:25 And then population standard deviation for florida is 8 cents.

03:34 And this is pretty much what we're saying is that when you're computing your margin of error, you're going to need the standard deviations and we're lucky enough to, well, actually you're going to need both the sample size and the standard deviation.

03:52 So the first requirement in part a was saying we want to get the point estimate, point estimate that differentiates the population mean gas gallon prices between these two.

04:32 States between these two states.

04:37 You know, that's what you're looking for.

04:39 You want to get the differences in gas prices between these two states.

04:44 And then the other thing you're saying is in part b, you want to get to 95 % confidence and the margin of error for making the point estimate prediction for the interval.

04:58 So margin of error.

04:59 In part c, we're saying, so at 95 % confident, what's the margin of error? that's part b.

05:10 And then in part c, and just recall that you're getting the difference between the two gas prices.

05:18 So x bar 1 minus x bar 2, and then you build an interval off of that.

05:27 In part c was saying the 95, we want to get the 95 % confidence interval that estimates, that estimates the difference in, i was going to say average, but i'm just going to say population means the difference in population means between california, and florida.

06:13 So that's what we're looking.

06:16 So we're going to jump right into it and in part a, m1 is the main gas price per gallon in california.

06:36 M2 is the main gas price per gallon in california.

06:36 M2 is the main gas price per gallon in florida.

06:47 Exba 1 as we said is, is, is, for the sample mean and this is this is population just remember that this is population the muse other population mean sample mean a gas price a gallon in california and x -2 is sample mean gas price per gallon in florida so we're looking at that and so the difference, you know, if you want to compute the difference, the point estimate difference, the point estimate difference becomes the point estimate difference for mu1 minus mu 2 is simply x 1 minus x bar 2, which is the same as $2 .04 minus $1 .14 minus $1 .1 .1 ,000, minus $1 .4 .5 .5 .000.

08:00 $0 .72 and that gives you $0 .32.

08:06 That's the difference.

08:08 And then in part b, we want to get the margin of error.

08:15 The confidence interval or the confidence level, the confidence level is given by 95 % confidence...

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