As an aid to the establishment of personnel requirements, the director of a hosp

Question

As an aid to the establishment of personnel requirements, the director of a hospital wishes to estimate the mean number of people who are admitted to the emergency room during a 24-hour period. The director randomly selects 64 different 24-hour periods and determines the number of admissions for each.

For this sample, sample mean = 19.8 and sample standard deviation = 5

Create a 95% Confidence Interval for the expected number of admissions for a 24-hour period.  

What formula will be used for the confidence interval?:

What is the degrees of freedom?:                   What is the t-coefficient?:

What is the sample mean?:

What is the sample standard deviation?:

What is the standard error?:

What is the Lower Bound of the Confidence Interval?:

What is the Upper Bound of the Confidence Interval?

What does the Confidence Interval Represent?:

Explanation / Answer

Answers

Formula

Let X = number of people who are admitted to the emergency room during a 24-hour period.

We assume X ~ N(µ, 2).

100(1 – ) % confidence interval for µ when 2 is unknown is: {Xbar ± (s/n)(t/2)}, where

Xbar = sample mean,

= population standard deviation,

s = sample standard deviation,

n = sample size and

t/2 = upper (/2) % point of t-Distribution with (n - 1) degrees of freedom..

t-coefficient (or what is popularly termed as critical t)

is the upper upper (/2) % point of t-Distribution with (n - 1) degrees of freedom

In the given question, = 5% and n = 64 and critical t = upper 2.5% point of t-distribution with degrees of freedom 63, = 1.998 [obtained using Excel Function – can also be directly read off from Standard Statistical Tables]

Degrees of freedom: is n – 1 = 64 – 1 = 63                  

Sample mean: = 19.8 (Xbar)

Sample standard deviation: 5 (s)

Standard error: s/n = 5/8

Substituting all the values in the formula, 95% Confidence Interval for mean number of people who are admitted to the emergency room during a 24-hour period is:

19.8 ± (5/8)(1.998) = 19.8 ± 1.249    

Lower Bound of the Confidence Interval: 18.551

Upper Bound of the Confidence Interval: 20.049

What does the Confidence Interval Represent?:

It represents the range within which the population mean is expected to lie with 95% confidence. In physical terms, CI = 19.8 ± 1.249 => there is just 5% chance that the population will lie below 18.551 or above 20.049

DONE

[Additional Inputs: If the population standard deviation is known, in the fomula,  just replace s by and t by Z]

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