Part 2 Inferring from the Sample

2.1 Getting Started

2.1.1 Research Question

No work needed here.

2.1.2 Understanding the Set-Up

The population in this context refers to all daily high temperatures in F for the months of January and February in Portland, Oregon. It is divided into two segments:

The sample consists of 1,482 daily high temperatures for the months of January and February in Portland, OR, covering the years 2000 to 2024. It is a Simple Random Sample, aimed at assessing the impact of historical temperatures (1970-1999) and estimating the recent average daily high t emperature through statistical analysis. The analysis will help to determine if the recent average significantly deviates from historical averages for this two-month period.

In summary, the variable of interest in this dataset is the daily high temperature (in F) for Portland, OR, during January and February from several years. This variable is quantitative, continuous, and significant for analyzing trends in temperature variations over time. By studying this variable, researchers can draw conclusions about temperature stability, monthly fluctuations, and changes in climate patterns in the region.

The parameter represents the mean (average) of the daily high temperatures (in degrees Fahrenheit) for all recorded days in January and February within the defined years.

The statistic (point estimate) of interest in the provided dataset is the average daily high temperature recorded for the months of January and February over the range of years from 2000 to 2024 in Portland, Oregon.

The value of the statistic (point estimate) for the average high temperature for January and February, rounded to two decimal places, is 45.25.

2.1.3 Procedures

The appropriate statistical procedures to use in this case would be t-procedures. This is because we do not know the population standard deviation, and we will rely on the sample data to calculate the standard deviation for the analysis.

Useage of t-procedures because:

The sample size is large (n30.)

The population standard deviation is unknown, and we must estimate it from the sample.

The t-distribution accommodates the added uncertainty introduced by estimating the standard deviation.

2.1.4 Wrap-Up

Given the goal of the analysis to determine whether the recent average daily high temperature in January and February (2000-2024) in Portland is significantly different from the historical average (1970-1999) and the fact that you're working with a sample of recent temperatures, here's how you should decide:

Use a one-sample t-test

Why t-test and not z-test? Comparing the sample mean (from the "recent" data sample) to a known historical population mean.

You do not know the population standard deviation for the recent period (you only have a sample of it).

The sample size is moderate (n < 100), so the Central Limit Theorem (CLT) doesn't fully justify using a z-test unless the population standard deviation is known which it's not.

2.2 Confidence Intervals

2.2.1 Research Question: Confidence Intervals

What is the true mean daily high temperature in Portland, Oregon, during January and February in recent years (2000-2024), and is it significantly different from the historical average (1970-1999)?

This question motivates estimating a confidence interval for the population mean of "recent" high temperatures so we can compare it to the known historical mean.

2.2.2 Confidence Interval

Degrees of Freedom:

To compute a confidence interval for the mean using a t-distribution, we use

Degrees of Freedom (df) = n1

We count the number of temperatures in the sample. From your list, there are 73 data points (you can confirm by counting the lines). So: df = 73 1 = 72

Critical Value (t*) for 98% Confidence

From a t-distribution table, for df = 72 and 98% CI: t* = 2.381

98% Confidence Interval

Sample Mean= 46.9041F

Sample size (n) = 73

sample standard deviation (5)=7,5465 F

t= critical value= 2.381

Confidence Interval = 46.9041 2.381X7.5465

73

= 46.9041 2.103

CI =(46.9041-2,103, 46.9041 +2:103)

CI = (44.8011, 49,0071)

(44.80F, 49.01 F)) (rounded)

We are 98% confident that the true mean daily high temperature for January and February in Portland, OR, from 2000 to 2024 is between 44.80F and 49.01F.

2.2.3 Wrap Up

There is statistical evidence at the 5% significance level to conclude that the average daily high temperature in January and February during recent years (2000-2024) is significantly different from the historical average of 45F.

2.3 Hypothesis Tests (Tests of Significance)

2.3.1 Research Question: Hypothesis Tests

Is the recent average high temperature (Jan.-Feb., 2000-2024) significantly different from the historical average of 48.35F?

2.3.2 Hypothesis Test

t = -1.637, p-value = 0.106, = 0.05 Fail to reject H

2.3.3 Wrap Up

There is not enough evidence at the 5% significance level to conclude that the recent average daily high temperature for January and February in Portland is significantly different from the historical average of 48.35F.

1) Based on your work in Part 2, what would you say about the daily high temperature for "recent" years? 2) This is a situation where the actual population value is known. The true mean high temperature is 48.68F . a) Is the confidence interval that you computed in 2.2 consistent with the actual mean? b) Is this consistent with the RESULTS of your hypothesis test in 2.3? Why or why not?