Confidence Interval for μ (raw data given)

To find the confidence interval and margin of error for the population mean when raw data is given, first find the sample mean, std and size using numpy.

Example: In a study of a new pain relieving drag, the measures of pain levels for 15 subjects are recorded below. Use the sample data to construct a 95% confidence interval for the mean pain levels for the population (assume normal).

8.6; 9.4; 7.9; 6.8; 8.3; 7.3; 9.2; 9.6; 8.7; 11.4; 10.3; 5.4; 8.1; 5.5; 6.9

Find the margin of error for 95% confidence interval

We will use the error_t_mean function above. But first we need the sample std and size.

x = [8.6, 9.4, 7.9, 6.8, 8.3, 7.3, 9.2, 9.6, 8.7, 11.4, 10.3, 5.4, 8.1, 
             5.5, 6.9]

import numpy as np
s = np.std(x,ddof=1)
n = np.size(x)

print('std',s)
print('size',n)

Show expected output

std 1.6722383060978339
size 15

error_t_mean(.95,s,n)  # values of s and n from above

Show expected output

Margin of Error is 0.9260547070143034

8.6; 9.4; 7.9; 6.8; 8.3; 7.3; 9.2; 9.6; 8.7; 11.4; 10.3; 5.4; 8.1; 5.5; 6.9

Find the 95% confidence interval for the mean

use the conf_t_mean function above but first find the mean, std and n.

x = [8.6, 9.4, 7.9, 6.8, 8.3, 7.3, 9.2, 9.6, 8.7, 11.4, 10.3, 5.4, 8.1, 
             5.5, 6.9]

import numpy as np
m = np.mean(x)
s = np.std(x,ddof=1)
n = np.size(x)

Call the function conf_t_mean with mean = m, std = s using results from the above code.

Try it in XYZ Crunch

Handing the summary statistics straight to the interval

The callout earlier in this section loads the 15 pain levels and reads their summary statistics off 1-Var Stats. This link takes the three numbers that came out - mean 8.226666666666667, Sx 1.6722383060978339, n = 15 - and hands them to the T Interval at C = 0.95, which is exactly what conf_t_mean does with m, s and n. It returns Lower Bound 7.300612, Upper Bound 9.152721, Margin of Error 0.926055 and df 14: the interval printed below, to every digit the panel shows. Doing it in two steps is the point - raw data becomes three numbers, and three numbers become an interval.

T Interval, x-bar = 8.226666666666667, s = 1.6722383060978339, n = 15, 95% confidence: 7.300612 to 9.152721

conf_t_mean(.95,m,s,n) # values of m,s and n from above

Show expected output

mean is 8.226666666666667 and std is 1.6722383060978339
( 7.300611959652363 , 9.15272137368097 )

Interpretation: $7.300611959652363 < \mu < 9.15272137368097$

Adapted from Python for Introductory Statistics, by Simon Aman (Truman College, City Colleges of Chicago), licensed under CC BY 4.0. Changes were made: reformatted as an accessible XYZ web edition with live in-browser code cells. License: CC-BY-4.0.