A blind taste test is conducted to determine which of two colas, Brand A or Brand B, individuals prefer. Individuals are randomly asked to drink one of the two types of cola first, followed by the ot…
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Question “A blind taste test is conducted to determine which of two colas, Brand A or Brand B, individuals prefer. Individuals are randomly asked to drink one of the two types of cola first, followed by the ot…”
A blind taste test is conducted to determine which of two colas, Brand A or Brand B, individuals prefer. Individuals are randomly asked to drink one of the two types of cola first, followed by the other cola, and then asked to disclose the drink they prefer. Results of the taste test indicate that 59 of 100 individuals prefer Brand A. Complete parts a through c. (a) Conduct a hypothesis test (preferably using technology) HO. p-p0 versus HI : pf p0 for Po-o48, 0.49, 0.50, , 0.68, 0.69, 0.70 at the α significance. For which values of po do you not reject the null hypothesis? What do each of the values of Po represent? 0.05 level of Do not reject the null hypothesis for the values of Po btenand, inclusively. Type integers or decimals as needed.) (b) Construct a 95% confidence interval for the proportion of individuals who prefer Brand A. The lower bound is The upper bound is (Round to three decimal places as needed.) (c) Suppose you changed the level of significance in conducting the hypothesis test to α=0.01. what would happen to the range of values for p0 for which the null hypothesis is not rejected? Why does this make sense? Choose the correct answer below. O A. O B. The range of values would increase because the corresponding confidence interval would increase in size The range of values would decrease because the corresponding confidence interval would decrease in size O C. The range of values would decrease because the corresponding confidence interval would increase in size ( D. The range of values would increase because the corresponding confidence interval would decrease in size
Answer
Part a
We will need to perform a z-test for population proportion.
H0: p = p0 versus H1: p p0
Where, p0 = .48, .49,…., .70
Z = (P- p)/sqrt(1)pq/n
Where, q = 1 + p
P is the sample proportion.
We are given
x = 59
n = 100
P = x/n = 59%/100 = 0.59
For p = .48, q = 1 – 0.48 = 0.52
Z = (0.59 – 0.48)/[0.48*0.52/100)
Z = 2.2018
P-value = 0.0277
(by using Excel)
P-value = a = 0.05
We reject the null hypothesis.
Let’s use excel to see how z tests can be run for other values of the p and 0.
p0 | Z | P-value | Take the decision |
0.48 | 2.2018 | 0.0277 | Reject |
0.49 | 2.0004 | 0.0455 | Reject |
0.50 | 1.8000 | 0.0719 | Reject not |
0.51 | 1.6003 | 0.1095 | Reject not |
0.52 | 1.4011 | 0.1612 | Reject not |
0.53 | 1.2022 | 0.2293 | Reject not |
0.54 | 1.0032 | 0.3158 | Reject not |
0.55 | 0.8040 | 0.4214 | Reject not |
0.56 | 0.6044 | 0.5456 | Reject not |
0.57 | 0.4040 | 0.6862 | Reject not |
0.58 | 0.2026 | 0.8394 | Reject not |
0.59 | 0.0000 | 1.0000 | Reject not |
0.60 | -0.2041 | 0.8383 | Reject not |
0.61 | -0.4100 | 0.6818 | Reject not |
0.62 | -0.6181 | 0.5365 | Reject not |
0.63 | -0.8285 | 0.4074 | Reject not |
0.64 | -1.0417 | 0.2976 | Reject not |
0.65 | -1.2579 | 0.2084 | Reject not |
0.66 | -1.4777 | 0.1395 | Reject not |
0.67 | -1.7014 | 0.0889 | Reject not |
0.68 | -1.9294 | 0.0537 | Reject not |
0.69 | -2.1622 | 0.0306 | Reject |
0.70 | -2.4004 | 0.0164 | Reject |
You should not reject the null hypothesis regarding the values of p and 0 that are between 0.50 to 0.68 inclusively.
Part b
Confidence interval for the Population Proportion
Confidence Interval = P +- Z* sqrt (P*(1-P)/n
P is the sample percentage, Z is the critical value and n is how large the sample is.
We are given
x = 59
n = 100
P = x/n = 59%/100 = 0.59
Confidence level = 95%
Z = 1.96
(by using z-table)
Confidence Interval = P +- Z* sqrt (P*(1-P)/n
Confidence Interval = 0.59 +- 1.96* sqrt (.59*(1-.59)/100).
Confidence Interval = 1.59 + 0.0492
Confidence Interval = 0.5964 +.59
Lower limit = .59 – 0.0964 = 0.494
Upper limit = .59 + 0.0964 = 0.686
Confidence interval = (0.494, 0.686).
Part c
Answer: A. Answer: A.
Explanation:
We know that the width or size of the confidence interval will increase if the significance level is decreased. The range of values will increase if the confidence interval is larger.
Conclusion
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