Question

a) An environmental conservation agency recently claimed that more than 30% of Canadian consumers have stopped buying a certain product because the manufacturing of the product pollutes the environment. You want to test this claim. To do so, you randomly select 980 Canadian consumers and find that 314 have stopped buying this product because of pollution concerns. At a = 0.05, can you support the agency’s claim?

*please round your p-hat to 4 decimals before substituting in the z-statistic formula*

b) Refer to question (b). Construct a confidence interval for the true proportion of Canadian consumers who have stopped buying the product at the following levels of confidence: i). 90% ii). 95%

Answer 1

• Here let the X be the random variable denoting the number of people who have stopped buying the product because of pollution concerns.
• The estimated proportion is =X/n=314/980=0.320
• The standard error is given by =0.01490
• HYPOTHESIS TO BE TESTED IS GIVEN BY
• Here let the null hypothesis be that the proportion of people who have stooped buying the product is equal to 0.3
• vs the alternate hypothesis be that the the proportion of people who have stooped buying the product is more than 0.3.
• TEST STATISTICS
• 
• by substituting the value test statistics obtained is
• 1.369
• CONCLUSION
• The critical value at 5% significant level is given by 1.645
• The test statistics obtained is less than 5% critical value.
• Hence we do not have enough evidence to reject the null hypothesis.
• Hence we conclude that the proportion of people who have stooped buying the product is equal to 0.3
• To calculate the 90% confidence interval
• The critical value is given by Za/2=Z0.1/2=Z0.05=1.645
• 0.95=(-1.645<N(0,1)<1.645)
• 0.95=(-1.645<<1.645)
• The margin of error is given by
• critical value * standard error
• 1.645*0.01490=0.0245
• Hence the interval is
• 0.3200.0245=(0.2950,0.3445)
• Hence the confidence interval is given by
• (0.2950<p<0.3445)
• To calculate the 95% confidence interval
• The critical value is given by Za/2=Z0.05/2=Z0.025=1.96
• 0.95=(-1.96<N(0,1)<1.96)
• 0.95=(-1.96<<1.96)
• The margin of error is given by
• critical value * standard error
• 1.96*0.01490=0.029204
• Hence the interval is
• 0.3200.029204=(0.290,0.3492)
• Hence the confidence interval is given by
• (0.290<p<0.3492)