# Chi-square goodness of fit test

### Chi-square goodness of fit test

#### Lessons

The chi-square distribution is the sum of standard normal distribution(s) squared. The degrees of freedom for a chi-square distribution is how many standard normal distribution(s) squared you are summing.

Normal distribution:

$X\sim N (\mu, \sigma^2)=$ Normal Distribution with mean ‘$\mu$’ and standard deviation ‘$\sigma$

So Chi-Square Distribution with k degrees of freedom:
$X^2=N_1(0,1)^2+N_2(0,1)^2+\cdots+N_k(0,1)^2$

Hypothesis Testing

Chi-Square distribution hypothesis testing comes in handy for seeing whether the observed value of some experiment fit the expected values.

$O_i$: the $i^{th}$ observed data point
$E_i$: the $i^{th}$ estimated data point

Test-Statistic:
$X^2=\frac{(O_1-E_1)}{E_1}+\frac{(O_2-E_2)}{E_2}+\cdots+\frac{(O_n-E_n)}{E_n}$

The critical value is found by looking at the Chi Distribution table
• Introduction
a)
Chi-Square Distributions

b)
Goodness of Fit Test (Hypothesis Testing with $X^2$)

• 1.
Determining Chi Square Distributions
If a $X^2$ distribution has 2 degrees of freedom then what is the area under this distribution that lies to the right of 5.99?

• 2.
If we have 12 squared standard normal distributions then what is the probability that their sum will be less than 6.304?

• 3.
Hypothesis Testing with the Chi Square Distribution
Emily is an avid potter. She pots at the UBC pottery club. The pottery club display some numbers representing the amount of pottery pieces their members produce in any given day. I go to the club and think that their estimate is incorrect, so I observe the amount of pottery produced by this studio throughout the week. Using the data given below, can I state with a significance level of $\alpha$=0.10 that the UBC pottery clubs has displayed incorrect numbers?
 Monday Tuesday Wednesday Thursday Friday Saturday Sunday Pottery Club: 15 20 15 25 10 25 30 My Observation: 13 18 19 22 12 23 24

• 4.
A car dealership claims that 20% of their cars sold are economy cars, 50% are family cars, 20% are luxury cars and the remaining 10% of cars sold are sports cars.
A list of their last 500 cars sold is: 115 economy cars, 270 family cars, 80 luxury cars and 35 sports cars.
With a significance level of $\alpha$=0.025 is the last 500 cars sold consistent with the car dealerships claim?