Infinite Population where sample size is greater than 30 and Standard Deviation of population is known/given:

Problem 1

Theatre owners in Tamil Nadu know that a hit film ran for an average of 84 days µ(µ) with a standard deviation of 10 days (σ) in each city. One film distributor wanted to compare the popularity of the film in his region with that of population parameter in Tamil Nadu. He selected 75 theatres in his region and found the popular move ran for 81.5 days(xbar)

  1. Choose appropriate hypothesis for testing if there is a significant difference in the distributor region and the population
  2. Significance level is 0.05 (5%)

Assumptions:

  1. Null hypothesis H0 = 84(mu) Population parameter
  2. Alternative hypothesis H1 not equal to 84 days (two-tailed test as H1≠84days)

Criteria:

  1. Significance level alpha = 0.05. for this z-tab is 1.96, Rejection area R|zcal| > R|ztable (1.96)|
  2. P_value must be greater than 0.05 for rejecting the Null hypothesis
  3. P_value must be less than or equal to 0.05 for accepting the Null Hypothesis

Test to be conducted:

 When Ho = <something> and H1≠ <something>need two-tailed test

Decision based on:

Criterion: z(cal) must be > z(table) for rejecting Null Hypothesis

sample size = 75 theatres

Specific value 81.5 xbar

Here sample size is >30

Population standard deviation (10 days) is given

Distribution could be z-distribution

Calculation of z

Since the absolute calculated value of z is 2.165 which is greater than Z critical value 1.96

we reject the null hypothesis H0 µ=84.

Using Python

Load libraries , read data and Calculate Z

Calculation of z Critical value and p-value:

OUTPUT/RESULT

Decision Making:

Decision whether Null hypothesis is to be rejected or accepted can be taken

 By Comparing the Calculated value and table value.                                   

  • if Calculated value is greater than  (>) z-table /critical value then Null hypothesis is rejected
  • If  calculated value is less than (<) z- table/critical value then Null hypothesis is accepted

By Finding out P-value corresponding to Calculated value

  • if p-value is greater than alpha value(0.05) then Null hypothesis is accepted
  • if p-value is less than alpha value(0.05) then Null hypothesis is  rejected and accept alternative hypothesis as difference is  statistically significant

There is a significant difference in the distributor region and the population mean.