I\'m not sure how to go about this question. Any help is welcome! Due a Ive Prob

Question

I'm not sure how to go about this question. Any help is welcome!

Due a Ive Problem: A fast food franchiser is considering building a restaurant in a now empty lot located at 35th Avenue and Thunderbird Road. The franchiser has had low revenues at its other two locations and needs larger revenues from the new site to compensate. An analyst determined that the site is acceptable only if the number of vehicles passing the location averages more than 500 per hour. State the Experiment, the uncertain Quantity, and the Possible values of the uncertain quantity. State the null and alternative hypotheses in words and in symbols. the definitions of Type I and Type Stat II Errors. Describe the Type I and Type Errors in terms of this problem Determine II the Relative Costs of Type 1 and Type II Errors. Choose wisely between a Type I error rate of alpha 0.10 or alpha w 0.01 (alpha 0.05 is not an option), and concisely justify your choice of alpha. A random sample of 61 hourly vehicle counts produced a sample mean of 511 vehicles per hour and a sample standard deviation s of 51 vehicles. Determine the test statistic. Determine the probability value for the test statistic. Provide a concise definition and interpretation of the probability value. Draw a complete normal distribution picture of this hypothesis test, show the calculated probability value, and explain the meaning of this calculated probability value. Determine if your analysis provides sufficient evidence to build at the designated location, including a clear conclusion and interpretation of the results.

Explanation / Answer

It is a question of testing an average (which is estimated as 511 after taking 61 hourly trials). The measurement was taken 61 times and the count could have been e.g. 520, 580, 440, 520...(around the range of 500) and the average of these 61 was 511. We have to test whether this average of 511 being greater than 500 is really possible or did it happen by chance.

The happening by chance is called p-value, after our calculation if our p-value turns to be less than alpha value (which is usually 5%) then we are fairly confident that our average is greater or lesser than the hypothesized value of 500 (greater or lesser is based on our hypothesis)

Let us assume mu1 = 511 (estimated). mu = hypothesied avg = 500

H0: NULL: mu1 = mu

ie. the estimated hourly count is equal to (or less than) the hypothesized hourly count of 500

H1: Alternate: mu1 > mu ie. mu1 > 500

ie. the estimated hourly count is greater than the hypothesized hourly count of 500

The uncertain quantity is the difference of mu1 and mu ie. mu1 - mu. It can take a) negative values in which case the hourly count is lower than the hypothesized 500

b) positive values in which case the the hourly count is greater than the hypothesized 500

c) zero in which case both of them are equal.

Type 1 error - Incorrect rejection of true null hypothesis. When the hypothesized is actually true, but the observed one turns out to be false. We make a type 1 error

Type 2 error - Incorrectly finding out a false null hypothesis. When the hypothesized is actually false, but the observed one turns out to be true. We make a type 2 error.

In our example, when the random hourly count 511 is less than or equal to 500 (meaning the null is true) and we find through our one sample z-test thatn 511 is greater than 500 (proving null is false), then we make a type 1 error

Similarly when the random hourly count 511 is greater than 500 (meaning the null is false) and we find through our one sample z-test that 511 is less than 500 (proving null is true), then we make a type 2 error.

Relative costs of the errors is - when we commit a type1 error, we are buiding a store when in reality it is going to fail. Because the null is true (null is 511 < 500) and we made a false observation 511 > 500 and thought it will be profitable.

When we commit a type2 error, we are not building a store when in reality it is going to be profitable. Because the null is false (null is 511 < 500) and we made an incorrection observation 511 < 500 and thought it will be useless to build a store.

it is the tradeoff between building costs of a store vs missed profit for the period of operations. If building cost is much much higher than the missed profit (which is usually the case), it is best to avoid type1 error. Hence type1 error should be very less, let us say keep it at 0.01

z-statistic = (mu1-mu)/ sqrt(s1^2/n)

= (511 - 500)/sqrt(51^2/61) = 1.6845

p-value = 1- norm.dist(1.6845,true) = 0.046 (since it is a right tailed test as we are testing for >hypothesized mean)

since p-value >0.01, there is more than a random chance of 511 greater than 500

which means fail to reject null hypothesis, meaning - null is true and hence the observed hourly count of 511 is less than the hypothesized hourly count of 500

DONOT build at the designated location

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