Introduction
The scenario I will be working with is that I am working at NCLEX Memorial Hospital in the infectious disease unit. As a healthcare professional, I need to work to improve the health of individuals, families and communities in various settings. The current situation that has posed as a problem at the hospital and raised eyebrows is that in the past few days, there has been an increase in patients admitted with a particular infectious disease. The basic statistical analysis shows that the disease does not affect minors hence the ages of the infected patients does play a critical role in the method that shall be required to treat the patients in order to impact positively on the health and well-being of the clients being served whether infected with the disease or associated with those infected. After speaking to the manager, we decided that we shall work together in utilising the available statistical analysis to look closer into the ages of the infected patients. To do that, I had to put together a spreadsheet with the data containing the information we shall need to carry out the analysis.
Data Analysis
From the data collected and input on an Excel sheet, there are sixty patients with the infectious disease. Of the patient’s whose data has already been collected an input on the excel sheet, the ages range from thirty-five years of age to seventy-six. There is only one patient in their thirties with the age of thirty-five. There are five patients in their forties, One forty-five, one forty-six, two at forty-eight and two at forty-nine. There are fifteen patients in their fifties, two at fifty, one fifty-two, one fifty-three, one fifty-four, four at fifty-five, one fifty-six, one at fifty-eight and four at fifty-nine. There are twenty-three patients in their sixties, five at sixty, one at sixty-two, one at sixty-three, two at sixty-four, one at sixty-five, three at sixty-eight and seven at sixty-nine. Finally, we have fifteen infected patients in their seventies, six at seventy, three at seventy-one, three at seventy-two, one at seventy-three, one at seventy-four and one at seventy-six. From the graph in Figure 1 below, the horizontal axis depicts the age group of patients infected with the disease and the vertical axis depicts the number of patients in the age group infected with the disease.
Figure 1
Data Classification
The qualitative variables in our data analysis would be the names of the patients infected with the disease while the quantitative data would be their ages, number of patients in each age category or age bracket that are infected with the disease and the number of patients in each specific age that are affected. The graph in Figure 1 above shows a quantitative analysis of the data. The discrete variables in this analysis are the number of patients infected with the disease because they could continue to increase to a finite number and we could still count them and add them to the analysis. Our continuous variable in this analysis is the age. For our analysis, we shall use the age in years. In our data set, the qualitative data has been omitted. The quantitative data is being measured based on the number of patients counted to have the disease and their ages. We have classified them in clusters of five in the graph in order to visualise the analysis. The discrete variable is being measured by the number of patients already diagnosed with the diseases and the continuous variable which is the age is currently being measured annually.
The Measures of Center and Variation
The measures of centre are the values in the middle of the data set which is the focal point. It can be determined using the mean medium and the mode. The mean defines the very centre and could also be defined as the average point. In our data analysis, it is important to figure out the centre of variation because it shall assist us to determine the most common age bracket that has been infected with the disease and shall therefore help us narrow down to the cause and effect faster by concentrating on the mean median and the mode of the data analysis.
The measures of variation are those that are utilised to describe data distribution and the variation between random variable. They show the range between the greatest and the least data values which are commonly known as the difference. Quartiles can be used to measure variation as they divide the data set into four equal parts. They are important as they assist in measuring probability of occurrence. In our case, they could be used with the most common age group to have the infectious disease and random variables such as their residents, their places of work and their activities or eating habits could be used to further analyse the data in order to figure out the source, the cure and the best way to prevent the spread. Arithmetically, it is derived by the variance and standard deviations of a data set.
Calculation of the Measures of Center and Measures of Variation
The Mean
The mean is the average of the data set and normally the centre of the data.
The Mean = Total of Ages / Sample Size
The Mean = 3705 / 60 = 61.81667
The Mean = 61.82
The Median
The Median = The Value at the Centre of the data when arranged according to the age magnitude. In our case that value is 61 (refer to the excel worksheet column G) as that is the 30th patient in that magnitude arrangement.
The Median = 61
The Midrange
The Midrange = The Midpoint is the sum of the lowest and the highest age values then divided by 2. In our data set, the lowest age value is 35 and the highest is 76
The Midrange = (35+76) /2
The Midrange = 111/2 =55.5
Midrange = 55.5
The Mode
The mode is the most frequent value in the data set. Our data set is composed of the ages of the infected patients with the disease. The most frequent age is 69 which has 7 patients
Mode= 69
The Range
The range of a data set is the difference between the highest and the lowest values in the set. Our data set is composed of the infected patient’s ages. The highest value is 76 and the lowest is 35.
The Range = 76 -35
The Range = 41
The Variance
Measures how far the data are from the mean. In this case the variance is
4698.9833/59 = 79.6437
The Standard deviation is calculated from the SQRT of the variance. In this case = 8.92
Conclusion
The conclusion from our study of the patients infected with the infectious disease in NCLEX Memorial Hospital is that they are currently sixty. The most infected patients range between the age of sixty and seventy-five but the highest number of infected patients are the age of sixty-nine as they are seven. The disease seems to be attained by the elderly from the age of thirty-five and seventy-six with the average age being sixty-one. Children, teenagers, the youth and the extremely elderly are not prone to the infectious disease.
Importance of constructing confidence intervals for the population mean
Confidence interval is a range of figures that provides an interval estimate of a set of unknown parameters (Heckard, Utts, & Utts, 2012). This is as opposed to using point estimation and contains the parameter’s value as well as stated probability.
Point estimate, on the other hand, uses a set of sample data to calculate a statistic (a single value) which serves as the best estimate of unknown parameter in a population whether random or fixed (Heckard, Utts, & Utts, 2012).
The best point for the population mean, E(X), is the sample mean, Xbar. By equating the population mean with the sample mean, we are solving for the parameters using the one-parameter case.
Confidence interval (C.I) is needed for bounding the mean and the standard deviation. In addition, the C.I will also be needed for obtaining the proportions, regression coefficients and the differences for the population proportions (Heckard, Utts, & Utts, 2012). C.I is also needed in obtaining and estimating the sampling error in relation to the parameter of interest.
Best point estimate for the population mean
The mean is the average of the data set and normally the centre of the data.
Sample Mean = Total of Ages / Sample Size
Sample Mean = 3705 / 60 = 61.81667
Sample Mean () = 61.82
The sample mean is () is the best point estimate of the population mean (µ).
The best point estimate for the population mean (µ) = 61.82
Confidence intervals for the population mean
Assuming that your data is normally distributed and the population standard deviation is unknown:
At 95% confident level:
C.I is given by:
With = 61.82, n = 60, s = 8.92, n-1 = 59
Margin of error =
C.I. = 61.82 1.9243 = (63.7443, 59.8957)
At 99% confident level:
With = 61.82, n = 60, s = 8.92, n-1 = 59
Margin of error =
C.I. = 61.82 3.0632 = (64.8832, 58.7568)
From the computations above, it can be seen that at 95% confidence level, the interval of the population mean lies between 58.7568 and 64.8832. The sample mean is 61.82 and therefore the mean lies within the interval of the figures. After increasing the confidence level to 99%, the interval also increases. At 99% confidence level, the sample mean is still within the interval as range of the interval figures is 58.7568 and 64.8832.
There are a number of observations that can be made by changing the confidence levels from 95% to 99%. First, the margin of error increase from 1.9243 to 3.0632. The critical value read off from the t-table also increases. The confidence interval also widens as a result of increasing the confidence interval form 95% to 99%. If the degree of confidence is increased, it affects the margin of error. The larger the confidence degree (level), the larger the margin of error (Heckard, Utts, & Utts, 2012). Since the confidence level is being increased from 95% to 99%, then the margin of error is also increased. This ultimately increases the confidence interval of the population mean.
The primary goal of statistics is to conduct a hypothesis. A hypothesis is a prediction about something; hypothesis testing is done to ascertain if a sampled proportion differs from a specified population. For the test to be valid eight steps are conducted to ensure the results are up to par (Lora M. and Richard J. Cook., 2009);
Step One -Identify and come up with a research question, this helps the researcher narrow down to what they want to test. For instance, is the number of patients admitted with infectious disease less than 65 years of age? Such questions are important as they help one in looking for the necessary data and conduct the test efficiently
Step Two-Ascertain that some expectations are met: The method of research used is Simple random sampling, the resultant outcome is only one, and the population is triple the sample size in question
Step Three-State the two types of hypothesis: Identify the null and alternative hypothesis. Null hypothesis shows equality while alternative does not.
Step Four-Determine a definite significant level that is the odds of refuting a null hypothesis through use of alpha
Step Five-Calculate the test statistic, this are constant values that are calculated from the available data when conducting a hypothesis test
Step Six-Change the test statistic into a P value; A p-value is the possibility that a selected sample would differ with the obtained one. It differs depending on the test used and is determined by use of the normal distribution table
Step Seven-Choose between the null and alternative hypothesis, this is where one has to determine whether the stated research question is correct. If the p-value is greater than the standardized value, the null hypothesis should be rejected
Step Eight-Creating a conclusion of your Research Question, determine whether or not the set values are sufficient evidence in confirming your research.
The p-value is the better approach as computation of one value is required to conduct the test, the critical approach is cumbersome as one has to compute the test statistic and also find the key value of the significance level
Option 2
Ho: p>=65; Ha p<65
The test is left tailed since the sample proportion is less than the hypothesized population proportion
The test statistics to be used is the t test since the standard deviation is unknown.
=-2.79
Degree of freedom is 60-1=59as observed from the t table the p- value is 0.05
0.5-0.05=0.45 critical value is -1.6
Subtracting alpha from the standard value of 0.5 then looking for the resultant difference in the z table.
Reject the null hypothesis since the test statistic is less than -1.6 which is the critical value.
There is sufficient evidence that the people who are admitted in NLEX hospital are less than 65 years of age.
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