constructing a one - proportion confidence interval\na recent poll of 750 randomly selected smartphone users…

constructing a one - proportion confidence interval\na recent poll of 750 randomly selected smartphone users found that 176 of the respondents admitted to walking into something or someone while texting on their cell phone. construct and interpret a 95% confidence interval for the proportion of all smartphone users who would admit to walking into something or someone while texting on their cell phone.\n2. plan. name the inference procedure you will use and check the conditions.\none - sample z - interval for p:\nrandom: random sample of 750 smartphone users\n10%: it is reasonable to assume that 750 is 10% of all smartphone users.\nlarge counts: the number of successes (176) and the number of failures (750 - 176 = 574) are both ≥

constructing a one - proportion confidence interval\na recent poll of 750 randomly selected smartphone users found that 176 of the respondents admitted to walking into something or someone while texting on their cell phone. construct and interpret a 95% confidence interval for the proportion of all smartphone users who would admit to walking into something or someone while texting on their cell phone.\n2. plan. name the inference procedure you will use and check the conditions.\none - sample z - interval for p:\nrandom: random sample of 750 smartphone users\n10%: it is reasonable to assume that 750 is 10% of all smartphone users.\nlarge counts: the number of successes (176) and the number of failures (750 - 176 = 574) are both ≥

Answer

Explanation:

Step1: Calculate sample proportion

Let $\hat{p}=\frac{x}{n}$, where $x = 176$ (number of successes) and $n=750$ (sample - size). So, $\hat{p}=\frac{176}{750}\approx0.235$.

Step2: Determine z - value for 95% confidence interval

For a 95% confidence interval, the critical value $z_{\alpha/2}=1.96$.

Step3: Calculate the margin of error

The formula for the margin of error $E = z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$. Substitute $\hat{p}=0.235$, $n = 750$, and $z_{\alpha/2}=1.96$ into the formula. First, calculate $1-\hat{p}=1 - 0.235 = 0.765$. Then, $\frac{\hat{p}(1 - \hat{p})}{n}=\frac{0.235\times0.765}{750}\approx\frac{0.179775}{750}\approx0.0002397$. And $\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\approx\sqrt{0.0002397}\approx0.0155$. So, $E=1.96\times0.0155\approx0.0304$.

Step4: Construct the confidence interval

The confidence interval is given by $\hat{p}-E<p<\hat{p} + E$. Substitute $\hat{p}=0.235$ and $E = 0.0304$ into the formula. We get $0.235-0.0304 < p<0.235 + 0.0304$, which simplifies to $0.205 < p<0.265$.

Step5: Interpret the confidence interval

We are 95% confident that the proportion of all smartphone users who would admit to walking into something or someone while texting on their cell - phone is between 0.205 and 0.265.

Answer:

The 95% confidence interval for the proportion of all smartphone users who would admit to walking into something or someone while texting on their cell - phone is $(0.205,0.265)$. We are 95% confident that the true proportion of such smartphone users lies within this interval.