Question

(Correct you answer to 4 decimal places and report the measurement unit when applicable)

Question (a) State which test (z-test, t-test, ANOVA, χ2 test) is most suitable for each of the following cases:

(i) A report claims that 60%, 30%, and 10% of primary kids go to school on foot, by school bus, and by public transport. Test, if the same ratio applied for secondary kids.

(ii) A survey has interviewed 20 kids. Each kid reported the amount of red-packet money he or she has received in year 2019 and year 2018. Test, if the amount of red-packet money received has been increased from year 2018 to year 2019.

(b) A mobile phone manufacturer has designed a new model of mobile phone and has invited 40 customers to take part in a trial session. After the trial session, the customers are invited to state their comments on the phone as well as to indicate how much they are willing to pay for this mobile phone. The results indicate that on the average a customer is willing to spend $5500 for it. Assume that the population standard deviation is the same as the old model, which is $1150. The director of the manufacturer suggests selling this new mobile phone at $6000. Test, at 3% significant level, whether the result collected from the trial session indicates the population mean amount a customer is willing to pay for this mobile phone is lower than the director’s suggestion.

(c) A recent research has been done on finding the relationship between alcohol content and the calories in 12-ounce beer.

Drink	Alochol Content (x)	Caloriies in 12 oz(Y)
Big Sky Lite	4.70%	163
Sierra Nevada Super	6.70%	215
MillerCoors X	8.10%	222
Anheuser Busch Zero	0.00%	70
MillerCoors Lite	4.15%	104
High Falls	5.10%	162
Sierra Nevada Dry	5.00%	158
Anheuser Busch Super	5.00%	155
Flying Dog	4.70%	158
Big Sky Extra	6.20%	159

(i) A researcher has studied the above figure and found that one of the drinks cannot be classified as beer and should be excluded from the following study. State the name of this drink.

(ii) Fit a regression line y = a + bx using the data above, exclude the drink stated in part (i), to predict the calories content in 12-ounce beer from its alcohol content.

(iii) Interpret the values of a and b in the regression line in part (ii).

(iv) Calculate the correlation coefficient between x and y, with the drink stated in part (i) is excluded. Comment on the relationship between the calories content in 12-ounce beer and its alcohol content.

(v) Estimate the calories content in 12-ounce beer when the alcohol content of an extra brand of drink is 6.50%

Answer 1

Solution

Part (a)

Sub-part (i)

χ2 test Answer 1

[3 levels of proportion of one group is being compared with 3 levels of proportion of another group]

Sub-part (ii)

t- test Answer 2

[one sample test for specific value of population mean when population standard deviation is unknown ]

Part (b)

Let X = amount ($) customers are willing to pay for the new mobile phone.

Let µ and σ be the mean and standard deviation of X.

Claim: The population mean amount a customer is willing to pay for this mobile phone is lower than the director’s suggestion.

Hypotheses:

Null H₀: µ = µ₀ = 6000 Vs Alternative H_A: µ < 6000

Test statistic:

Z = (√n)(Xbar - µ₀)/σ = -2.7498,

where

n = sample size;

Xbar = sample average;

σ = known population standard deviation.

Summary of Excel Calculations is given below:

n	40
µ0	6000
σ	1150
Xbar	5500
Zcal	-2.7498
α	0.03
Zcrit	-1.8808
p-value	0.002982

Distribution, Significance level α, Critical Value and p-value

Under H₀, Z ~ N(0, 1)

Critical value = lower α% point of N(0, 1).

p-value = P(Z < Zcal)

Using Excel Functions, Statistical NORMINV   NORMSDIST Zcrit and p-value are found to be as shown in the above table.

Decision:

Since Zcal < Zcrit, or equivalently, since p-value < α. H₀ is rejected.

Conclusion:

There is sufficient evidence to support the claim and hence we conclude that the population mean amount a customer is willing to pay for this mobile phone is lower than the director’s suggestion. Answer 3

Part (c)

Sub-part (i)

Miller Coors X cannot be classified as beer Answer 4

[Excluding this, the average alcohol content is 4.61% and 8.1% is much higher]

Sub-part (ii)

Regression line

y = 61.0547 + 19.1272 Answer 4

Sub-part (iii)

Interpretation of the values of a and b

When the alcohol % is zero, the calories is expected to be 61.0547. Answer 5

For every additional point in alcohol percentage, the calories is expected to increase by 19.1272. Answer 6

Sub-part (iv)

Correlation coefficient between x and y = 0.8922 Answer 7

Comment on the relationship between the calories content in 12-ounce beer and its alcohol content.

Correlation coefficient of 0.8922 indicates that there strong positive linear relationship. Further, r-square being 0.7960 implies that 79.6% of variation in calorie valus is influenced by alcohol percentage. Answer 8

Sub-part vi)

Estimate of the calories content in 12-ounce beer when the alcohol content of an extra brand of drink is 6.50% is 185.3459 calories Answer 9

Details of calculations

n	9
Xbar	4.6167
ybar	149.3333
Sxx	28.92
Syy	13284
Sxy	553
β1cap	19.12172
β0cap	61.05474873
r	0.892198633
r^2	0.796018401

DONE