Question

A survey​ asked, "How many tattoos do you currently have on your​ body?" Of the 12311231 males​ surveyed, 190190 responded that they had at least one tattoo. Of the 10671067 females​ surveyed, 143143 responded that they had at least one tattoo. Construct a 9090​% confidence interval to judge whether the proportion of males that have at least one tattoo differs significantly from the proportion of females that have at least one tattoo. Interpret the interval. Let p 1p1 represent the proportion of males with tattoos and p 2p2 represent the proportion of females with tattoos. Find the 9090​% confidence interval for p 1 minus p 2p1−p2.

Answer 1

Answer:

$(0.154-0.134) - 1.64 \sqrt{\frac{0.154(1-0.154)}{1231} +\frac{0.134(1-0.134)}{1067}}=-0.00402$  

$(0.154-0.134) + 1.64 \sqrt{\frac{0.154(1-0.154)}{1231} +\frac{0.134(1-0.134)}{1067}}=0.044$  

We are confident at 95% that the difference between the two proportions is $-0.00402 \leq p_1 -p_2 \leq 0.044$  

Since the confidence interval contains the value 0 we can conclude that at 10% of significance we don't have enough evidence to conclude that the true proportions for female and male with tattos differs

Step-by-step explanation:

Information given

$p_1$ represent the real population proportion of males with tattoos

$\hat p_1 =\frac{190}{1231}=0.154$ represent the estimated proportion of males with tattos

$n_1=1231$ is the sample size for males

$p_2$ represent the real population proportion of female with tatto

$\hat p_2 =\frac{143}{1067}=0.134$ represent the estimated proportion of females with tattos

$n_2=1067$ is the sample size of female

$z$ represent the critical value

Confidence intrval

The confidence interval for the difference of two proportions would be given by this formula  

$(\hat p_1 -\hat p_2) \pm z_{\alpha/2} \sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1} +\frac{\hat p_2 (1-\hat p_2)}{n_2}}$  

For the 90% confidence interval the value of $\alpha=1-0.90=0.1$ and $\alpha/2=0.05$, and the critical value for this case would be:  

$z_{\alpha/2}=1.64$  

Replacing the info given we got:

$(0.154-0.134) - 1.64 \sqrt{\frac{0.154(1-0.154)}{1231} +\frac{0.134(1-0.134)}{1067}}=-0.00402$  

$(0.154-0.134) + 1.64 \sqrt{\frac{0.154(1-0.154)}{1231} +\frac{0.134(1-0.134)}{1067}}=0.044$  

We are confident at 95% that the difference between the two proportions is $-0.00402 \leq p_1 -p_2 \leq 0.044$  

Since the confidence interval contains the value 0 we can conclude that at 10% of significance we don't have enough evidence to conclude that the true proportions for female and male with tattos differs