How to Calculate Confidence Intervals (With Examples)

In statistics, a confidence interval (CI) is an estimate of an interval that likely would contain an unknown population parameter. The concept was introduced in 1937 by Jerzy Neyman, a Polish mathematician, and statistician. In statistics, the concept of CI is used as a measure of uncertainty. The CI is often expressed as a percentage, and the 90%, 95%, and 99% are the CI that are used frequently. These percentages reflect the confidence level of a given population parameter. In this paper, an article that has employed the concept of CI will be identified and summarized. The paper will then discuss the context in which the concept of CI is used in the article. 

Summary of the Study 

The study that was identified is the 2019 study by Righy et al. titled “Prevalence of post-traumatic stress disorder (PTSD) symptoms in adult critical care survivors: a systematic review and a meta-analysis.” The aim of the study was to evaluate the prevalence of PTSD symptoms in adults who have been discharged from the intensive care unit (ICU). To achieve its objectives, the study conducted a comprehensive systematic review and meta-analysis. In particular, the researchers conducted a comprehensive literature search using a wide range of databases, including MEDLINE, EMBASE, LILACs, Web of Service, PsycNET, and Scopus. From their literature search, the researchers retrieved a total of 13,267 studies. However, only 48 of the studies retrieved were reviewed. 

From their systematic review and meta-analysis, the researchers found that the overall prevalence of PTSD symptoms in adult critical patients was 19.83% (95% CI, 16.72-23.13; I 2 =90%). Various studies reported difference prevalence rates. Righy et al. (2019) found the point prevalence estimates to be 15.93% “(95% CI, 11.15-21.35; I 2 =90%; 17 studies), 16.80% (95% CI, 13.74-20.09; I 2 =66%; 13 studies), 18.96% (95% CI, 14.28-24.12, I 2 =92%; 13 studies), and 20.21% (95% CI, 13.79-27.44; I 2 =58%; 7 studies)” at 3, 6, 12, and >12 months after discharge, respectively (1). 

The researchers concluded that PTSD symptoms are highly likely to affect 1 in every 5 adults who are discharged from ICUs, with a high expected prevalence 1 year after discharge. For this reason, the researchers state that adult critical care survivors ought to be screened for PTSD symptoms. In addition, the researchers recommend that ICU survivors be cared for due to the negative impact of PSTD on health-related quality of life. Lastly, the researchers conclude that the casual relationships between ICU stay and PTSD should be further explored in order to prevent PTSD in ICU survivors. 

The context in which the Concept of Confidence Interval is used in the Study 

Righy et al. (2019) employed the concept of CI in their study. In particular, the researchers the concept of CI to determine the prevalence of PTSD symptoms in ICU survivors. The researchers used a sample size of 48 and 95% CI to measure the range of values that would likely contain the parameter they are investigating, which is the prevalence of PTSD in adult critical care survivors. Righy et al. (2019) presented their results in forest plots with 95% Cis or scatter plots with point estimates and 95% CI. They used the 95% CI to perform their analyses. In the study context, the confidence interval means the range of adult critical care survivors exposed to PTSD symptoms after being discharged from ICU. According to Righy et al. (2019), the overall prevalence of PTSD symptoms in adult care critical survivors is 19.83% (95% CI, 16.72-23.13; I 2 =90%). 

Week 7 Discussion: Rejection Region 

Statistical inference refers to the process of drawing conclusions from a given sample data. This includes testing hypotheses and deriving estimates. In other words, after reviewing data from a given sample, one can make inferences about the general population using the sample population. In this paper, a sample data will be located and retrieved from the Internet. The paper will then identify the trend in the data and make inferences about the data. The paper will also outline the steps needed to complete the hypothesis test. 

The data that will be used for analysis in this paper was obtained from the Federal Reserve Bank of St. Louis website, which is located at https://fred.stlouisfed.org/series/CPRPTT02GBA661N . The data is about the consumer price index and the retail price index of all items from 1975 to 2017 for the United Kingdom (UK). However, it is important to note that the data excludes the mortgage interest rate. In this paper, the data retrieved from the Federal Reserve Bank of St. Louis will be used for analysis. Table 1 shows the data retrieved. 

Table 1: Consumer Price Index: Retail Price Index 

Frequency: Annual
observation_date	CPRPTT02GBA661N
1975-01-01	15.5649501307569
1976-01-01	18.2805044627435
1977-01-01	21.1872814812359
1978-01-01	23.0043774620521
1979-01-01	25.8977016282275
1980-01-01	30.2698786312492
1981-01-01	33.9578820021496
1982-01-01	36.8492843322797
1983-01-01	38.7615111973375
1984-01-01	40.4853981299574
1985-01-01	42.5773166652493
1986-01-01	44.1215119276352
1987-01-01	45.7736202057998
1988-01-01	47.8727782974743
1989-01-01	50.6791393826006
1990-01-01	54.8026192703461
1991-01-01	58.5070159027128
1992-01-01	61.2572497661366
1993-01-01	63.0869971936389
1994-01-01	64.5874649204864
1995-01-01	66.4172123479888
1996-01-01	68.3741814780168
1997-01-01	70.2563143124415
1998-01-01	72.1272217025257
1999-01-01	73.7773620205800
2000-01-01	75.3115060804490
2001-01-01	76.9167446211413
2002-01-01	78.6155285313377
2003-01-01	80.8231992516370
2004-01-01	82.6080449017774
2005-01-01	84.4826941066417
2006-01-01	86.9560336763330
2007-01-01	89.7773620205800
2008-01-01	93.6014967259121
2009-01-01	95.4536950420954
2010-01-01	100.0000000000000
2011-01-01	105.2871842843780
2012-01-01	108.6735266604300
2013-01-01	111.9925163704400
2014-01-01	114.7277829747430
2015-01-01	115.9064546304960
2016-01-01	118.0767072029930
2017-01-01	122.5706267539760

Source: The Federal Reserve Bank of St. Loius 

By looking at the data, one can observe an important trend. The consumer price index and the retail price index has been increasing steadily from 1975 to 2017. From 1975 to 2017, the consumer price index and retail price index increased from 15.5650 to 122.5706. This trend can be determined graphically by plotting the observation date, and the consumer price index and retail price index on a scatter plot. Figure 1 shows the consumer price index and retail price index from 1975 to 2017 for the UK. 

Figure 1: Consumer Price Index: Retail Price Index 

From the graph, it is clear that the consumer price index and the retail price index of all consumer goods less mortgage interest rates in the UK have been increasing from 1975 to 2017. 

Based on this trend and the history of the data set retrieved, one can make the claim that the consumer price index and the retail price index for all items less mortgage interest rate in the UK will increase in the future. To predict the increase, one can use regression analysis. In particular, one can use the following equation to determine the consumer price index and the retail price index for a given year: 

Since the consumer price index and the retail price index for 2017 is known (122.57), one can use this value to make a claim. For example, one can claim that the consumer price index and the retail price index for 2020 will be greater than 122.57. 

i.e. 

To test this claim, one can use the right-tailed test. A right-tailed test is a test that is used in statistics to test a hypothesis. The hypothesis usually contains the greater (>) sign symbol. In this case, the right-tailed test can be used to test the claim that the consumer price index and the retail price index for all items in 2020 will be greater than that of 2017. 

To test this claim, there are several steps needed to complete the hypothesis test. The steps are outlined below: 

Write the hypothesis statement. 

Use the z-score formula to calculate the test statistic: 

Choose an alpha level 

In this case, the standard alpha level (0.05) will be used. 

Use the z-table to test the hypothesis. 

Using the z-table, look up for the value that corresponds to the z-score and the confidence level. 

WK 8 Discussion: Regression 

Regression is a statistical method used to determine the strength and character of the relationship between two or more variables. In this case, one of the variables is a dependent variable, and the other is an independent variable. In this paper, sample data will be retrieved from the Internet. The data will be divided into two pairs (x, y); one will be the dependent variable, and the other will be the independent variable. The paper will then delve into analyzing the data to determine the type and strength of the correlation between these two variables. 

The data that was used for analysis in this paper was retrieved from Excel Jet, which is available at https://exceljet.net/formula/bmi-calculation-formula . Table 2 shows the data retrieved. The data pertains to the height and weight of individuals. Excel Jet used this data to calculate the body mass index (BMI), a crude population-level measure for obesity. However, this data will be used in this paper to determine the type and strength of the correlation between height and weight. 

Table 2: Height vs. Weight 

Height (m)	Weight (kg)
1.88	74.84
1.83	83.91
1.63	61.23
1.78	79.38
1.8	102.06
1.68	54.43
1.68	65.77
1.85	115.67

Source: Excel Jet (n.d). 

An independent variable is a variable that stands alone; it is not affected by other variables. In this case, the height of the individual is the independent variable. On the contrary, a dependent variable is a variable that changes. In this case, the weight of the individual is the dependent variable. 

By just looking at the data, I expect to see a moderate correlation between height and weight. This is because I can see some individuals have greater weight values from the data yet they are short. I also expect the r-squared vale to be roughly between 30 and 60%. This will indicate that there is a moderate variation between the two variables. To determine the type and strength of the correlation, one can plot the values of these two variables (Height and Weight) on a scatter plot. Figure 2 shows a graphical presentation of these two variables. 

Figure 2: Height vs. Weight 

Just like predicted, there is a moderate correlation between height and weight. Also, the r-squared is 0.4969 (49.69%), indicating that the proportion of variation between the dependent and independent variables is moderate. The equation of the regression line and the r-squared is shown below: 

The equation of the regression can be used to estimate the weight of other individuals as long as their heights are known. For example, if the height of an individual is 1.9m, his or her weight will be: 

References 

Excel Jet. (n.d). BMI calculation formula. https://exceljet.net/formula/bmi-calculation-formula 

Federal Reserve Bank of St. Louis. (2017). Consumer price index: retail price index: All items less mortgage interest rate for the United Kingdom. https://fred.stlouisfed.org/series/CPRPTT02GBA661N 

Righy, C., Rosa, R. G., da Silva, R. T. A., Kochhann, R., Migliavaca, C. B., Robinson, C. C., ... & Falavigna, M. (2019). Prevalence of post-traumatic stress disorder symptoms in adult critical care survivors: a systematic review and meta-analysis.  Critical Care ,  23 (1), 213.