Question

You take your family on a wonderful, relaxing vacation to the beach. About 15 minutes after you’ve settled into the perfect spot in the sand, your oldest child tells you he’s bored. To keep him busy you tell him to collect some shells, because you read online that the beach where you’re staying is known for having lots of different colors of shells wash up on the beach. A few days later he’s collected over 500 shells, and he tallies up how many of each color he has in the table below. You’re curious if his collection of shells has the same distribution of colors as the overall beach has, so you go online and find the distribution of shell color percentages you should expect to find at that particular beach, and add those to your data table. Perform a hypothesis test to determine if the color distribution of your son’s seashell collection is what you’d expect at that beach. Use α = 0.10.

Step 1) What type of hypothesis test is required here?

               

Step 2) Verify all assumptions required for this test.

Step 3) State the null and alternate hypotheses for this test using correct symbols and notation.

Step 4) Fill in the table of expected values below. Round each value to 2 decimal places.

	White	Red	Black	Orange	Blue	Other	Total
Expected # of Shells

Run the correct test in MINITAB and provide the information below. Use correct symbols and round answers to 3 decimal places.

               

Test Statistic
Degrees of freedom      

Critical Value
p-value

               

Step 5) State your statistical decision and justify it.

Step 6) Interpret your decision within the context of the problem: what is your conclusion?

	White	Red	Black	Orange	Blue	Other	Total
# of Shells	309	46	73	45	31	8
Expected percentages	57%	12%	14%	8%	6%	3%	100%

Answer 1

	White	Red	Black	Orange	Blue	Other	Total
# of Shells	309	46	73	45	31	8	512
Expected percentages	57%	12%	14%	8%	6%	3%	100%
expected number	291.84	61.44	71.68	40.96	30.72	15.36

Expected number of 1 colour = Expected % of the colour * total number of shells

Step 1) What type of hypothesis test is required here?

This test is chi-square test for goodness of fit.

We want to check whether our observed data matches the previously expected data,if the given model is suitable for our data.

Step 2) Verify all assumptions required for this test.

• The sample is selected at random- The kid does not show any preference for any colour therefore we can say that data is random.
• The variable to be studied should be categorical . - We are studying whether all the beaches have same colour distribution of sea shells.Our category is colour.
• Expected number should be greater than 5 for all the observations. - As we can from the above table all the exp values are greater than 5.

All the assumptions are satisfied.

Step 3) State the null and alternate hypotheses for this test using correct symbols and notation.

: The model is a good fit for our data. No significant diff between observed and exp values.

VS

: The model is not a good fit for our data. There is a significant diff between observed and exp values.

Step 4) Fill in the table of expected values below. Round each value to 2 decimal places.

	White	Red	Black	Orange	Blue	Other	Total
# of Shells	309	46	73	45	31	8	512
Expected percentages	57%	12%	14%	8%	6%	3%	100%
expected number	291.84	61.44	71.68	40.96	30.72	15.36
Ei-Oi	17.16	-15.44	1.32	4.04	0.28	-7.36
	294.466	238.394	1.742	16.322	0.078	54.170
	1.009	3.880	0.024	0.398	0.003	3.527	8.841

Test Statistic= = 8.841
Degrees of freedom = 5 (df = no. of obs - 1 = 6 - 1)

Critical Value = =16.75   

p-value = 0.1156 p-value = P ( > T.S.)

Step 5) State your statistical decision and justify it.

Decision criteria: Reject null hypothesis if

Test Stat > Critical value

OR

p - value >

Decision: In our case

Test stat (8.841) < Critical Value (16.75) and p-value(0.1156) > 0.01

Therefore, we do not reject the null hypothesis at 0.01 level of significance.

Step 6) Interpret your decision within the context of the problem: what is your conclusion?

Conclusion: The model is a good fit for our data. The proportion of the coloured cells is same for all the beaches.