Question

There are 4 parts to this assignment. Your answers to each of these parts must be hand-written (preferably in pen) and include a diagram or diagrams and any calculations similar to the hand-written examples.

1. One bank company determines the number of ATM machines they put in their branches based on the percentage of their customers who would use their machines. Currently one of their branches has 1 ATM machine and they will only put in another machine if the percentage of their customers using their existing machine is greater than 35%. A random sample of 120 customers entering this branch revealed that 48 used its ATM machine. At the .10 level of significance can this bank conclude that more than 35% of all its customers use the machine? Should they put in another machine?

2. Do Windsorites like their fried chicken to be spicy or not spicy? Currently, a restaurant which specializes in fried chicken only makes non-spicy chicken. It is now considering offering spicy chicken to its customers but, only if it can be convinced that more than 30% of its chicken eating customers would prefer spicy chicken. Over a one-week period, it prepared non-spicy chicken and a limited amount of spicy chicken in order to conduct an experiment. During this period of its offer, it randomly selected 60 of its chicken ordering customers and gave them pieces of both spicy and non-spicy chicken at no cost. Twenty-seven of these randomly selected customers stated that they preferred the spicy chicken. At the .01 level of significance, should the restaurant start offering spicy chicken to its customers?

To answer all of the above questions:

• state the two hypotheses
• calculate the value of the test statistic, z
• calculate the p-value of this test (diagram needed for p-value)
• determine the critical value of z (diagram needed)
• compare the value of z with the critical value of z and compare the p-value with α and state whether or not the null hypothesis should be rejected and write a concluding statement (diagram needed [critical value of z/rejection region])

Note: Please hand written and draw a box around the final answer, Thank you.

Answer 1

a) Null hypothesis : The proportion of customers using the existing ATM (P) 0.35

Alternative hypothesis : The Proportion of customers using the existing ATM (P) > 0.35

Test statistic is given by -

where, n is the sample size = 120

p is the Sample Proportion of customers using the existing ATM = 48/120 = 0.4

P is the specified value of the population proportion under the null hypothesis = 0.35

Hence, the value of test statistic will be -

= 1.15

Now, the critical value of z at 0.10 level of significance for upper tailed test is nearly 1.28

It can be obtained from the z table by finding the z value corresponding to the area of 0.90.

Since, z calculated (test statistic) < critical value of z, that is, the value of the test statistic lies in the acceptance region, hence, the null hypothesis may not be rejected. Hence, the proportion of customers using the existing ATM is less than and equal to 0.35, hence, they should not put in the new ATM.

P value = P( z > 1.15)

=1 - P( z < 1.15)

= 1 - 0.8741

= 0.1251

The P(z < 1.15) is the area under the z curve on the left side of z = 1.15 and equal to 0.8741.

It can be obtained from the z table by finding the area corresponding to the z value = 1.15

Now, since, p value (0.125) > level of significance (0.10), the null hypothesis may not be rejected at 0.10 level of significance suggesting that proportion of customers using the existing ATM is less than 0.35, hence, the new atm should not be put in.

b) Null hypothesis : The proportion of chicken eating customers who prefer spicy chicken (P) 0.30.

Alternative hypothesis : The Proportion of chicken eating customers who prefer spicy chicken (P) > 0.30

Test statistic is given by -

where, n is the sample size = 60

p is the sample proportion of chicken eating customers who prefer spicy chicken = 27/60 = 0.45

P is the hypothesized value of the population proportion = 0.30

Hence, the value of the test statistic will be -

  

= 2.534

The critical value of z statistic at 0.01 level of significance for right tailed test is 2.33 as obtained from the z table by finding the z value corresponding to the area close to 0.99.

Since, the value of the test statistic > critical value of z, i.e, the value of test statistic falls in the rejection region, the null hypothesis may be rejected. Hence, the proportion of chicken eating customers who prefer spicy chicken is more than 0.30, so, restaurant should start offering spicy chicken to it's customers.

P value = P(z > 2.534) = 1 - P(z < 2.534)

The P(z < 2.534) is the area under the z curve on the left side of z = 2.53. It can be obtained from the z table by finding the area corresponding to the z value 2.53 and is equal to 0.9943

Hence, the P value = 1 - 0.9943

= 0.0057

Since, p value (0.0057) < level of significance (0.01), Hence, we have strong evidence against the null hypothesis, so, null hypothesis must be rejected, suggesting that chicken eating customers prefer spicy chicken more.

So, restaurant should srart offering spicy chicken to it's customers.