Mathematics, 02.07.2020 20:01 adelawilliams60

A restaurant supply store surveys all chefs in the area and finds that 64 of the 85 head chefs in the area prefer their stores to the other restaurant supply store in town. To increase their sales, they are doing a statistical study of all chefs in the area. Use a calculator to find above what proportion will 80% of all sample proportions be in the sampling distribution of sample proportions of size 45.

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Answer from: ballin3294

80% of all sample proportions in the sampling distribution of sample proportions of size 45 will be above 0.6988.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

When the distribution is normal, we use the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$ , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$ .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean $\mu = p$ and standard deviation $s = \sqrt{\frac{p(1-p)}{n}}$

In this question:

$p = \frac{64}{85} = 0.7529, n = 45, \mu = 0.7529, s = \sqrt{\frac{0.7529*0.2471}{45}} = 0.0643$

Above what proportion will 80% of all sample proportions be in the sampling distribution of sample proportions of size 45.

Above the 100 - 80 = 20th percentile, which is X when Z has a pvalue of 0.2. So X when Z = -0.842.

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$-0.842 = \frac{X - 0.7529}{0.0643}$

X - 0.7529 = -0.842*0.0643

X = 0.6988

80% of all sample proportions in the sampling distribution of sample proportions of size 45 will be above 0.6988.

Answer from: Quest

the answer is n

Answer from: Quest

function 2 has a greater rate of change than function 1

function 2 has a greater y-intercept than function 1

step-by-step explanation:

$\newline{\text{the rate of change is change per unit time}}$ $\newline{\text{for (x,y), rate of change is }\frac{y_2-y_1}{x_2-x_1} \text{ or change in y divided by change in x}}$ $\newline{\text{for function 1, the cost goes up by 75-45=30 each time every hour}}$ $\newline{\text{that means that the slope of function 1 is 30 per hour}}$ $\newline{\text{for the 2nd function, 40 is added every hour. therefor, the slope is 40 per hour}}$ $\bigskip\newline{\text{since 40 per hour> 30 per hour, function 2 has a greater rate of change than function 1}}$ $\bigskip\newline{\text{the y intercept is the y value when x is 0}}$ $\newline{\text{for function 1, the y value when x is 0 is 45}}$ $\newline{\text{for function 2, the y value when x is 0 (or the flat rate) is 50}}$ $\newline{\text{since 50> 45, function 2 has a greater y intercept than funciton 1}}$ $\bigskip\bigskip\newline{\text{the answers are: }}$ $\newline{\text{ function 2 has a greater rate of change than function 1}}$ $\newline{\text{function 2 has a greater y-intercept than function 1.}}$

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