It is known that for a certain firm specializing in manufacturing female T-shirts, the ‘Small’ sizes have a shoulder size of 14 inches or less. If these T-shirt shoulder sizes are normally distributed with a mean of 16 inches and a standard deviation of 2 inches
(i)What proportion of manufactured T-shirt sizes will be ‘Small’?
(ii) In a random sample of 40 T-shirts manufactured by this firm, what is the probability that none will be ‘Small’?
(iii) What size of T-shirt will be greater than 80% of all the sizes produced?
To solve these questions, we'll be using the concept of the normal distribution and its properties. The normal distribution is a bell-shaped curve that is symmetric and characterized by its mean and standard deviation.
(i) To find the proportion of T-shirt sizes that will be 'Small', we need to calculate the cumulative probability up to the cutoff of 14 inches. In other words, we'll calculate the area under the normal distribution curve from negative infinity up to 14 inches.
To do this, we can use a standard normal distribution table or a statistical software. However, since the mean (μ) and standard deviation (σ) are given, we can calculate the z-value (standardized score) using the formula:
z = (x - μ) / σ
where x is the cutoff value, μ is the mean, and σ is the standard deviation.
In this case, we have:
z = (14 - 16) / 2
z = -1
Looking up the standard normal distribution table or using software, we find that the cumulative probability at z = -1 is approximately 0.1587. This means that about 15.87% of the T-shirt sizes will be 'Small'.
(ii) To find the probability that none of the 40 T-shirts in a random sample are 'Small', we need to calculate the cumulative probability of all T-shirts being larger than the cutoff (14 inches).
Let's calculate the z-value for 14 inches again:
z = (14 - 16) / 2
z = -1
Since we want the probability that all 40 T-shirts are larger than 14 inches, we raise this cumulative probability to the power of 40:
P(no T-shirt is 'Small') = (1 - 0.1587)^40
P(no T-shirt is 'Small') ≈ 0.1237
So, the probability that none of the 40 T-shirts will be 'Small' is approximately 0.1237 or 12.37%.
(iii) To find the T-shirt size that will be greater than 80% of all the sizes produced, we need to find the value on the distribution curve corresponding to the cumulative probability of 80%.
Again, we can use the standard normal distribution table or a statistical software to find the z-value for a cumulative probability of 80%.
Using software or consulting the table, we find that the z-value for a cumulative probability of 80% is approximately 0.845.
Now, we can rearrange and solve the z-score formula to find the corresponding T-shirt size:
z = (x - μ) / σ
Rearranging, we get:
x = z * σ + μ
x = 0.845 * 2 + 16
x ≈ 17.69
Therefore, a T-shirt size of approximately 17.69 inches will be greater than 80% of all the T-shirt sizes produced.