Project #4 - Introduction to Statistical Inference

Purpose – Are North Shore Students Different?

In this project, students will demonstrate their understanding of the normal distribution, sampling distributions, confidence intervals and hypothesis tests to determine if NSCC students differ from typical averages or if any differences are just due to random variation.

Question 1: Sample v. Population

Tasks:

Load and store the sample NSCC Student Dataset using the read.csv() function.

# Store the NSCC student dataset in environment
nscc_student_data_1_ <- read.csv("/Users/patrickmannion/Downloads/nscc_student_data (1).csv")

Find the sample mean and sample size of the PulseRate variable in this dataset and answer the question that follows below.

# Mean pulse rate of this sample
mean(nscc_student_data_1_$PulseRate,na=TRUE)

## [1] 73.47368

# Find the sample size of pulse rates (hint: its how many non-NA values are there)
sample(nscc_student_data_1_$PulseRate)

##  [1] 50 66 92 74 80 96 60 66 66 64 NA 66 70 72 69 80 62 60 65 92 NA 92 71 88 65
## [26] 72 87 89 98 85 75 80 60 80 64 89 70 61 60 56

The average pulse rate of the sample, n/a values are counted as zeros. The sample size of pulse rates is 38, and the mean is approximately 73.474 Questions:

Do you expect the sample mean to equal the true population mean? Why? The sample mean will vary slighlty from the true population mean since the sample size is part of the total population, and if all possible sample means are taken, they equate to the population mean.
If we took a different sample, would we get the same results? Results will not be the same with a different sample.

Question 2: Confidence Intervals

Task: Construct 90%, 95%, and 99% Confidence Intervals for the mean pulse rate of all NSCC students. Assume that σ = 14.

# Store mean
mean<-mean(nscc_student_data_1_$PulseRate,na=TRUE)

# Calculate lower bound of 95% CI
mean-1.96*(14/sqrt(38))

## [1] 69.02233

# Calculate upper bound of 95% CI
mean+1.96*(14/sqrt(38))

## [1] 77.92504

# Calculate 90% Confidence Intervals 
mean-2.58*(14/sqrt(38))

## [1] 67.61425

mean+2.58*(14/sqrt(38))

## [1] 79.33312

# Calculate 99% Confidence Intervals
mean-1.645*(14/sqrt(38))

## [1] 69.73772

mean+1.645*(14/sqrt(38))

## [1] 77.20964

The lower and upper bounds of the 95%, 90% and 99% confidence interval are (69.022, 77.925), (67.614, 79.333), and (69.738, 77.210),calculated using the mean pulse rate, known values for 90, 95, and 99% confidenc intervals, and the given std. deviation of 14 Questions:

Interpret your 95% CI in plain language It can be assumed with 95% certainty that if we choos a random pulse rate value from the sample, it will be between the values of 69.022 and 77.925
How does the interval change as confidence increases?

79.333-67.614

## [1] 11.719

77.925-69.022

## [1] 8.903

77.210-69.738

## [1] 7.472

Subtracting the upper bound from the lower bound gives the range between the two values. As confidence intervals increase, the range between the lower and upper bounds decreases - Which interval would you report and why? I would report the 95% confidence interval, From our data it provides the greatest possible coverage of the data, as it has the greatest possibel range of values. It is also common industry standard to use the 95% interval, and is something to bcome familiar with.

Question 3: Hypothesis Testing with a Confidence Interval

Consider the national average pulse rate for US adults to be 72 bpm. Let’s test the claim that NSCC students differ from that national average.

\(H_0: \mu = 72\)
\(H_A: \mu \neq 72\)

Tasks:

Use your confidence interval – Does it contain 72?
The confidence interval does contain 72
Based on that – Do you reject or fail to reject the null hypothesis?
Based on the evidence provided, we fail to reject the null hypothesis. Questions:
What does your result suggest about NSCC students?
It suggests that NSCC students have a pulse rate that is within the expectations of the national average.
Does “fail to reject” mean NSCC students are the same as average?
No, it means it falls within a statistically acceptable range of values seen as “average”.

Question 4: Hypothesis Testing with a P-value

Task: Recall the sample data you got in question 1. For the hypotheses in question 3, compute the test statistic of that sample data and the p-value using pnorm().

sd(nscc_student_data_1_$PulseRate, na=TRUE)

## [1] 12.51105

pnorm(72, 73.474, 12.511 )

## [1] 0.4531066

The p-value is calculated using the sample mean, the quantity being measured, and the standard deviation calculated above. It is approximately 0.453.

Questions:

What does your p-value represent in context?
P-values represent the probability of obtaining our sample data by random chance, given that nothing is different between the sample we’re using and the comparable population.
Using an α = 0.05, is your conclusion the same as with the 95% confidence interval? If not, why might they differ?
0.453>0.05, we fail to reject the null hypothesis

Question 5: Reflection

If you repeated this study of collecting NSCC students’ pulse rates to determine if they differ from the national average:

#- Could your conclusions change? Why?
Yes, conclusions could change. This sample only supplies a small portion of the school’s entire population, it may not refelct the entire student body. #- What assumptions did you rely on in using these methods?
It wss assumed that the data was sampled randomly, has a normal distribution, and is appropriately spread and scaled. #- Are there any limitations, flaws, questions, or concerns you have with the analysis done in this project?
May be more useful to take multple pulse rates at different activity levels.