a farmer cuts hazel twigs to make into bean poles to sell at the market ,he says that the sticks are each 240cm long but in the lengths of the sticks are normally distributed such that 55% of the sticks are longer than240cm and 10% are longer than 250cm.find the probability that a randomly selected stick is shorter than 235cm
To find the probability that a randomly selected stick is shorter than 235cm, we'll need to use the concept of the standard normal distribution.
First, let's convert the lengths of the sticks into a standard normal distribution. We'll use the z-score formula for this, which is:
Z = (X - μ) / σ
where:
Z = the z-score
X = the value we want to convert
μ = the mean of the distribution
σ = the standard deviation of the distribution
We are given that 55% of the sticks are longer than 240cm. This means that 45% (100% - 55%) of the sticks are shorter than or equal to 240cm. We can assume that this 45% is distributed symmetrically on both sides of 240cm.
Next, we know that 10% of the sticks are longer than 250cm. Therefore, 90% (100% - 10%) are shorter than or equal to 250cm. Again, we can assume that this 90% is distributed symmetrically on both sides of 250cm.
Now, let's calculate the z-score for 240cm and 250cm.
For 240cm:
Z = (240 - μ) / σ
For 250cm:
Z = (250 - μ) / σ
Since we have two equations and two unknowns (μ and σ), we can solve for both.
From the first equation, we know that Z when X = 240 is 0.45 (45%):
0.45 = (240 - μ) / σ
From the second equation, we know that Z when X = 250 is 0.9 (90%):
0.9 = (250 - μ) / σ
Now, we have a system of equations that we can solve simultaneously. We can rearrange the first equation to find μ in terms of σ:
240 - μ = 0.45σ
μ = 240 - 0.45σ
Substituting this expression for μ into the second equation:
0.9 = (250 - (240 - 0.45σ)) / σ
Simplifying the equation:
0.9 = (250 - 240 + 0.45σ) / σ
0.9 = (10 + 0.45σ) / σ
Cross-multiplying:
0.9σ = 10 + 0.45σ
Simplifying further:
0.9σ - 0.45σ = 10
0.45σ = 10
σ = 10 / 0.45
σ ≈ 22.22
Now that we have found the value of σ, we can substitute it back into one of the earlier equations to solve for μ:
0.45 = (240 - μ) / 22.22
Cross-multiplying:
22.22 * 0.45 = 240 - μ
9.999 = 240 - μ
μ = 240 - 9.999
μ ≈ 230
Finally, we have the values of μ (approximately 230) and σ (approximately 22.22).
To find the probability of a randomly selected stick being shorter than 235cm, we need to find the z-score for 235cm using the formula:
Z = (X - μ) / σ
Z = (235 - 230) / 22.22
Z ≈ 0.2253
Now, we can use a standard normal distribution table or a calculator to find the probability associated with Z = 0.2253. This probability represents the likelihood of a stick being shorter than 235cm.
The probability that a randomly selected stick is shorter than 235cm is approximately equal to the value from the standard normal distribution table or calculator for Z = 0.2253.