Mathematics, 14.04.2020 17:27 melissapulido198

(A) Construct a 95% confidence interval for the average width of an elevator rail. (B) How large a sample of elevator rails would we have to measure to ensure that we could estimate, with 95% confidence, the average diameter of an elevator rail within 0.01 inch

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Answer from: lupitasgirl3326

(a) The 95% confidence interval for the average width of an elevator rail is (2.048, 2.052).

(b) The sample size required is 4.

Step-by-step explanation:

The width of 100 elevator rails have been measured. The sample mean and sample standard deviation of the elevator rails are 2.05 inches and 0.01 inches respectively.

According to the Central Limit Theorem if we have a population with mean μ and standard deviation σ and we take appropriately huge random samples (n ≥ 30) from the population with replacement, then the distribution of the sample mean will be approximately normally distributed.

Then, the mean of the distribution of sample means is given by,

$\mu_{\bar x}=\mu$

And the standard deviation of the distribution of sample means is given by,

$\sigma_{\bar x}=\frac{\sigma}{\sqrt{n}}$

Since the sample size is quite large the population standard deviation can be estimated by the sample standard deviation.

So, $\sigma_{\bar x}=\frac{s}{\sqrt{n}}=\frac{0.01}{\sqrt{100}}=0.001$

(A)

The (1 - α)% confidence interval for population mean (μ) is:

$CI=\bar x\pm z_{\alpha/2}\times \sigma_{\bar x}$

The critical value of z for 95% confidence level is:

$z_{\alpha/2}=z_{0.05/2}=z_{0.025}=1.96$

Compute the 95% confidence interval for the average width of an elevator rail as follows:

$CI=\bar x\pm z_{\alpha/2}\times \sigma_{\bar x}$

$=2.05\pm 1.96\times 0.001\\=2.05\pm 0.00196\\=(2.04804, 2.05196)\\\approx(2.048, 2.052)$

Thus, the 95% confidence interval for the average width of an elevator rail is (2.048, 2.052).

(B)

The margin of error of a (1 - α)% confidence interval for population mean (μ) is:

$MOE=z_{\alpha/2}\times \frac{s}{\sqrt{n}}$

Given:

MOE = 0.01

Compute the sample size as follows:

$MOE=z_{\alpha/2}\times \frac{s}{\sqrt{n}}$

$n=[\frac{z_{\alpha/2}\times s }{MOE}]^{2}$

$=[\frac{1.96\times 0.01}{0.01}]^{2}$

$=3.8416\\\approx 4$

Thus, the sample size required is 4.

Answer from: Quest

lockers 1, 4, 9, 16, 25, 40, and 49 are open

i ran through it by the other answer is wrong

Answer from: Quest

the population of other specie of birds will decline or other specie of birds will find some alternate source of food to survive.

explanation:

according to the competitive exclusion principle, which is also known as gause's law, two specie living in the same niche and feeding on the same resource cannot co-exist stably.

as we have well studied the theory of natural selection which states that only those organisms survive which have better tendency to compete for food and adapt to the environmental conditions.

coming towards the question, what wi ll happen when one species outcompetes the other for the small fish , the other specie will not get enough food and will start to die or decline in population. eventually either other specie needs to find another alternative food source for itself or migrate at some other location where there is less competition otherwise it will just not exist anymore at that niche.

hope it !

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