X is the number of hours a patient waits in the emergency room before being called back to be examined. X−π- is the mean wait time of 70 patients in the emergency room.
The error bound would become EBM = 0.245. This error bound decreases because as sample sizes increase, variability decreases and we need less interval length to capture the true mean.
The sample size needed would increase. As the confidence level increases, απΌ decreases and z(a2)π§(π2) increases. To maintain the same error bound, the size of the sample needs to increase.
X is the number of “successes” where the woman makes the majority of the purchasing decisions for the household. P′ is the percentage of households sampled where the woman makes the majority of the purchasing decisions for the household.
The error bound would increase. Assuming all other variables are kept constant, as the confidence level increases, the area under the curve corresponding to the confidence level becomes larger, which creates a wider interval and thus a larger error.
X is the time in minutes it takes to complete the U.S. Census short form. X−π- is the mean time it took a sample of 200 people to complete the U.S. Census short form.
The level of confidence would decrease because decreasing n makes the confidence interval wider, so at the same error bound, the confidence level decreases.
The interval is greater because the level of confidence increased. If the only change made in the analysis is a change in confidence level, then all we are doing is changing how much area is being calculated for the normal distribution. Therefore, a larger confidence level results in larger areas and larger intervals.
The error bound for the mean would decrease because as the CL decreases, you need less area under the normal curve (which translates into a smaller interval) to capture the true population mean.
We estimate with 96% confidence that the mean amount of money raised by all Leadership PACs during the 2011–2012 election cycle lies between $47,292.57 and $456,415.89.
Yes, the intervals (0.72, 0.82) and (0.65, 0.76) overlap, and the intervals (0.65, 0.76) and (0.60, 0.72) overlap.
We can say that there does not appear to be a significant difference between the proportion of Asian adults who say that their families would welcome a White person into their families and the proportion of Asian adults who say that their families would welcome a Latino person into their families.
We can say that there is a significant difference between the proportion of Asian adults who say that their families would welcome a White person into their families and the proportion of Asian adults who say that their families would welcome a Black person into their families.
X = the number of adult Americans who feel that crime is the main problem; P′ = the proportion of adult Americans who feel that crime is the main problem
Since we are estimating a proportion, given P′ = 0.2 and n = 1000, the distribution we should use is N(0.2,(0.2)(0.8)1000−−−−−−√)π(0.2,(0.2)(0.8)1000).
CI: (0.18, 0.22)
Check student’s solution.
One way to lower the sampling error is to increase the sample size.
The stated “± 3%” represents the maximum error bound. This means that those doing the study are reporting a maximum error of 3%. Thus, they estimate the percentage of adult Americans who feel that crime is the main problem to be between 18% and 22%.
No, the confidence interval includes values less than or equal to 0.50. It is possible that less than half of the population believe this.
CL = 0.75, so α = 1 – 0.75 = 0.25 and α2=0.125 zα2=1.150πΌ2=0.125 π§πΌ2=1.150. (The area to the right of this z is 0.125, so the area to the left is 1 – 0.125 = 0.875.)
EBP=(1.150)0.52(0.48)1,000−−−−−−−√≈0.018πΈπ΅π=(1.150)0.52(0.48)1,000≈0.018
(p′ - EBP, p′ + EBP) = (0.52 – 0.018, 0.52 + 0.018) = (0.502, 0.538)
Yes – this interval does not fall less than 0.50 so we can conclude that at least half of all American adults believe that major sports programs corrupt education – but we do so with only 75% confidence.
X is the height of a male Swede, and x––π₯_ is the mean height from a sample of 48 male Swedes.
Normal. We know the standard deviation for the population, and the sample size is greater than 30.
CI: (70.151, 71.85)
Figure 8.21
The confidence interval will decrease in size, because the sample size increased. Recall, when all factors remain unchanged, an increase in sample size decreases variability. Thus, we do not need as large an interval to capture the true population mean.
X is the time needed to complete an individual tax form. X¯¯¯π¯ is the mean time to complete tax forms from a sample of 100 customers.
N(23.6,7100√)π(23.6,7100) because we know sigma.
(22.228, 24.972)
Figure 8.22
It will need to change the sample size. The firm needs to determine what the confidence level should be, then apply the error bound formula to determine the necessary sample size.
The confidence level would increase as a result of a larger interval. Smaller sample sizes result in more variability. To capture the true population mean, we need to have a larger interval.
According to the error bound formula, the firm needs to survey 206 people. Since we increase the confidence level, we need to increase either our error bound or the sample size.
We estimate with 95% confidence that the mean amount of contributions received from all individuals by House candidates is between $287,109 and $850,637.