Question

D3 a product is either rectangular or triangular. What is the probability that, in a random sample of 10 units, there are more than 8 rectangular units of product? Round your answer to 4 decimal places. Assume that the population proportion of rectangular products is the same as the sample proportion of rectangular products

Product weight (g)	Product height (mm)	Product colour	Product shape
18	53	Yellow	Rectangular
11	93	Yellow	Rectangular
19	43	Red	Rectangular
15	69	Red	Rectangular
19	42	Yellow	Rectangular
11	78	Red	Triangular
11	74	Yellow	Rectangular
20	90	Yellow	Rectangular
11	90	Blue	Rectangular
11	84	Yellow	Triangular
19	48	Red	Triangular
13	56	Blue	Triangular
20	79	Blue	Rectangular
11	60	Red	Rectangular
13	53	Blue	Rectangular
16	93	Yellow	Rectangular
16	91	Yellow	Triangular
20	67	Yellow	Rectangular
19	63	Blue	Triangular
15	53	Yellow	Rectangular
12	75	Blue	Triangular
18	91	Red	Rectangular
35	91	Red	Triangular
25	93	Red	Rectangular
30	44	Yellow	Rectangular

Answer 1

The Given data is

Product weight (g)		Product height (mm)	Product colour		Product shape
18		53	Yellow		Rectangular
11		93	Yellow		Rectangular
19		43	Red		Rectangular
15		69	Red		Rectangular
19		42	Yellow		Rectangular
11		78	Red		Triangular
11		74	Yellow		Rectangular
20		90	Yellow		Rectangular
11		90	Blue		Rectangular
11		84	Yellow		Triangular
19		48	Red		Triangular
13		56	Blue		Triangular
20		79	Blue		Rectangular
11		60	Red		Rectangular
13		53	Blue		Rectangular
16		93	Yellow		Rectangular
16		91	Yellow		Triangular
20		67	Yellow		Rectangular
19		63	Blue		Triangular
15		53	Yellow		Rectangular
12		75	Blue		Triangular
18		91	Red		Rectangular
35		91	Red		Triangular
25		93	Red		Rectangular
30		44	Yellow		Rectangular

Also we know that the population proportion of rectangular products is the same as the sample proportion of rectangular products

let p be the sample proportion of rectangular products and P be the population proportion of rectangular products , the p is same as P or unbiased estimate of population parameter

Now consider p be the probability of success . Here success denoted proportion of rectangular products in given sample.

Now we are given samples of size 25 .

In sample of size 25 we have 17 recatangle products and 8 triangle products .

Thus p = 17 / 25 = 0.68

So getting an rectangular unit is binomial process with probability of success p = 0.68

i.e   P (X = x ) = f(x) ~ B( n , p = 0.68 )

                  P(X=x) =  

We need to find the probability that, in a random sample of 10 units, there are more than 8 rectangular units of product

Thus here n = 10 , p =0.68   We need to find P(X > 8 )

Now P(X>8) = P(X=9) + P(X=10)

                     = +

                     = 10 * + 1 *

                     = 0.09947872 + 0.02113923 = 0.1206179

Thus P(X>8) = 0.1206179

Hence probability that , in a random sample of 10 units , there are more than 8 rectangular of product is 0.1206179