Question

1. The holiday season is always the busiest for e-commerce businesses, with the most sales and biggest discounts. One research claims that there were more online shoppers n 2017 compared to 2016.  In 2017, 800 customers were asked “Did you shop online during the holiday season?” 584 replied “yes” and 216 replied “no”. In 2016, 600 customers were asked “Did you shop online during the holiday season?”  423 replied “yes” and 177 replied “no”. Test the claim at the significant level α=0.01.
   1. State the null hypothesis and the alternative hypothesis. Make sure you clearly indicate the parameters for “year 2017” or “year 2016”:              

   2. H0:

                    Ha:

   3. Find the sample proportion for 2017, the sample proportion for 2016, and the grand proportion.
   4. Calculate the test statistic. Specify which test statistic you are finding (“z” or “t”).
   6. Find the p-value using the appropriate table. Be sure to include any appropriate degrees of freedom.   Draw the curve showing p-value, critical value(s), and rejection region.
   8. What is your decision and interpret the decision. Make sure to include the weight of the evidence, the conclusion, and the significance level.
   10. What is the 80% confidence interval of the difference in percentages of online shopping for 2017 and 2016? Interpret your result(s).

Answer 1

(a)

H0: Null Hypothesis: p1 p2 ( there were not more online shoppers n 2017 compared to 2016)

(b)

HA: Alternative Hypothesis: p1 > p2 ( there were more online shoppers n 2017 compared to 2016) (Claim)

(c)

the sample proportion for 2017 is given by:

1 = 584/800 = 0.73

(d)

the sample proportion for 2016 is given by:

2 = 423/600 = 0.705

(e)

the grand proportion. is given by:'

(f)

the test statistic. is given by:

So,

Z_Obt = 1.03

(g)

We are finding Z Test Statistic.

(h)

By Technology,

p - value = 0.1515

Since we are using Z statistic, degrees of freedom is not considered.

(i)

From Table, critical value of Z_C = 2.33

Rejection Region:

Reject Region:

Reject H0 if Z > 2.33

Drawing the curve showing p - value, critical value and Rejection Region is as follows:

Title: Z Test for Two Proportions: Z_Obt = 1.03,   p - value= 0.1515

Step 1: Draw the Standard Normal Curve showing mid value = 0 with x axis ranging from - 3.5 to + 3.5

Step 2: Draw a vertical line at x = 1.03 on RHS of mid value from bottom up to cutting the curve

Step 3: Shade the entire area of the curve to the right of the vertical lane drawn.

Step 4: Mark the shaded area as Rejection Region

(j)

Since p - value = 0.1515 is greater than = 0.01, the difference is not significant. Fail to reject null hypothesis.

Conclusion:

The data support the claim that there were more online shoppers n 2017 compared to 2016.at the significance level = = 0.01

(k)

n1 = 800

1 = = 584/800 = 0.73

n2 = 600

2 = 423/600 = 0.705

= 0.20

80%Confidence Interval:

= (- 0.006, 0.056)

So,

Answer is:

(- 0.006, 0.056)

Interpretation:

Since 80% Confidence interval includes 0, we conclude at 80% confidence that there were no significant difference between online shoppers n 2017 and 2016