Question

More shoppers do the majority of their grocery shopping on Saturday than any other day of the week. However, is the day of the week a person does the majority of grocery shopping dependent on age? A study cross-classified grocery shoppers by age and major shopping day. The data as follows UMUR/AGE Atas 55 35 54 HARI MEMBELI BELAH Bawah 35 UTAMA /MAJOR SHOPPING DAY Over 55 Under 35 24 56 Sabtu Saturda 48 176 144 Hari selain hari Sabtu/A day other than Saturda 152 a) Dapatkan jangkaan frekuensi bagi semua sel Find the expected frequencies for all the cells (4 markah/marks) b) Adakah terdapat bukti perbezaan yang beerti di antara umur dengan hari membeli-belah utama barangan runcit pada aras keyakinan 0.05? Is there evidence of a significant difference among the age groups with respect to major grocery shopping day at 0.05 significance level? (4 markah/marks) Terdapat perbezaan yang beerti di antara umur 35 54 dan atas 55, dan di antara usia bawah 35 dan atas 55 apabila prosedur Marascuilo digunakan. There is a significance difference between the 35-54 and over 55 groups, and between the under 35 and over 55 groups when the Marascuilo procedure was used c) Bagaimana kedai-kedai runcit boleh menggunakan maklumat ini dan di (b) untuk meningkatkan pemasaran dan jualan? Terangkan secara khusus How can grocery stores use this information and in (b) to improve merketing and sales? Be specific (2 markah/marks)

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Answer 1

Answer: - Date: --17/5/2019 H1: Not all proportions are equal 48+56+24 The pooled proportion -p- - .2133 200+200+200 E11 200(.2133)- 42.66, E12- 200(.2133)-42.66, E13 200(.2133) 42.66 E21-200(1.2133)-157.34, E22-200(1-1-2133)-157.34, E23-200(1-1-2133)-157.34 Note that all the expected frequencies (Eij) are greater than 5 (48-42.66)2 (56-42.66)2 (24-42.66)2, (152-157.34)2(144-157.34)2(176-157.34)2 - 42.66 16.525 42.66 42.66 157.34 157.34 157.34 The critical value is 2os with k-1 3-1-2 degrees of freedom is 5.99 Since 16.525 > 5.99 there is sufficient evidence at 5% significance level to argue that not all age groups prefer Saturday shopping at the same proportions After the Chi-square test is run, if we conclude not all the proportions are equal, sometimes you need to have some more insight about the way the proportions differ (which one is the largest etc.). When observing the results we can rank the sample proportions from the smallest (age group of "Over 54" with a sample proportion of 24/200 .12 to the largest (age group of 35 54 with a sample proportion of 56/200= .28. Now, since we have already determined that not all the real proportions are the same based on the test results, one can argue that people of age group 35- 54 tend to shop on Saturdays more than people of the "Over 54" group

After performing this procedure we'll be able to rank the different proportions individually or at least into subgroups. First we build a critical range. Than we calculate the absolute value of the difference between every pair of sample proportions. For each pair where the absolute difference mentioned is greater than the critical range, the two corresponding (real) proportions are considered different. To calculate the critical range for the comparison of pi and pj let us define pi- -aP)P-P12 Critical range(ij)- For the pair of sample proportions (j) if Ipi-Py| > Critical Range (j) then at α% significance level pi # pj. 12 48 56 р,-200-24, p,-20 0-.28, p,-200-06, x2.05 with 3-1-2 degrees of freedom-5.99 Critical ranges and testing Critical range(1,2)- 5.99 (2424)28(1-28) 200.107 Since.04 <.107 there is insufficient evidence to infer that pı is different than p2 ricalrangel1 3 5.09- 84 .24(1-.24) .06(1-.06) lp! _ pal=.18 critical range(1,3)=15.99 * 200 200

Since .18> .084 there is sufficient evidence to infer that there is a difference between pi and p Moreover, since the sample proportion associated with the customers under the age of 35 is greater than the proportion of those over 54, we can conclude at 5% significance level that the younger group tends to shop on Saturday more than the older group 15.99 *( lPa-Pal = .22 Critical range(2,3)- .088 200 200 Since .22 > .088 there is sufficient evidence to infer that there is a difference between p2 and p3. Moreover, since the sample proportion associated with the customers in the age group of 35-54 is greater than the proportion of those over 54, we can conclude at 5% significance level that the younger group tends to shop on Saturday more than the older group.