Arrow down to n and enter Out of a random sample of 65 freshmen at State University, 31 students have declared a major. In a group of 50 teens, 13 reported having more than friends on Facebook. Arrow down to x and enter The Berkman Center Study referenced in Figure talked to teens in smaller focus groups, but also interviewed additional teens over the phone.
When the study was complete, teens had answered the question about their Facebook friends with saying that they have more than friends. Compare the results to those in Figure.
If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size. Try other products: 0. The largest possible product gives us the largest n. To calculate the sample size n , use the formula and make the substitutions. Round the answer to the next higher value. Suppose an internet marketing company wants to determine the current percentage of customers who click on ads on their smartphones.
Jensen, Tom. Prince Survey Research Associates International. Saad, Lydia. The Field Poll. Some statistical measures, like many survey questions, measure qualitative rather than quantitative data. In this case, the population parameter being estimated is a proportion. It is possible to create a confidence interval for the true population proportion following procedures similar to those used in creating confidence intervals for population means.
The formulas are slightly different, but they follow the same reasoning. Then the confidence interval for a population proportion is given by the following formula:. Simply imagine four additional trials in the study; two are successes and two are failures. Calculate , and proceed to find the confidence interval. When sample sizes are small, this method has been demonstrated to provide more accurate confidence intervals than the standard formula used for larger samples.
Use the normal distribution for a single population proportion. Use the following information to answer the next two exercises: Marketing companies are interested in knowing the population percent of women who make the majority of household purchasing decisions.
It would decrease, because the z-score would decrease, which reducing the numerator and lowering the number. Use the following information to answer the next five exercises: Suppose the marketing company did do a survey.
They randomly surveyed households and found that in of them, the woman made the majority of the purchasing decisions. We are interested in the population proportion of households where women make the majority of the purchasing decisions. State the confidence interval, sketch the graph, and calculate the error bound. List two difficulties the company might have in obtaining random results, if this survey were done by email.
Use the following information to answer the next five exercises: Of 1, randomly selected adults, identified themselves as manual laborers, identified themselves as non-manual wage earners, identified themselves as mid-level managers, and identified themselves as executives.
Suppose we want to lower the sampling error. What is one way to accomplish that? Use the following information to answer the next five exercises: A poll of 1, voters asked what the most significant issue was in the upcoming election.
Sixty-five percent answered the economy. We are interested in the population proportion of voters who feel the economy is the most important. Use the following information to answer the next 16 exercises: The Ice Chalet offers dozens of different beginning ice-skating classes.
All of the class names are put into a bucket. The 5 P. The procedure to find the confidence interval, the sample size, the error bound , and the confidence level for a proportion is similar to that for the population mean, but the formulas are different. How do you know you are dealing with a proportion problem? First, the underlying distribution is a binomial distribution. There is no mention of a mean or average. To form a proportion, take X , the random variable for the number of successes and divide it by n , the number of trials or the sample size.
When n is large and p is not close to zero or one, we can use the normal distribution to approximate the binomial. Recall that a proportion as the number of successes divided by n.
EBP is error bound for the proportion. For the normal distribution of proportions, the z -score formula is as follows. Suppose that a market research firm is hired to estimate the percent of adults living in a large city who have cell phones.
Five hundred randomly selected adult residents in this city are surveyed to determine whether they have cell phones. Of the people surveyed, responded yes — they own cell phones. X is binomial. Remember that the area to the right of z 0. This can also be found using appropriate commands on other calculators, using a computer, or using a Standard Normal probability table. Ninety-five percent of the confidence intervals constructed in this way would contain the true value for the population proportion of all adult residents of this city who have cell phones.
Arrow down to APropZint. Arrow down to and enter Arrow down to C-Level and enter. The confidence interval is 0. Suppose randomly selected people are surveyed to determine if they own a tablet. Of the surveyed, 98 reported owning a tablet. A state public health department wishes to investigate the effectiveness of a campaign against smoking. In a survey commissioned by the public health department, of 1, randomly selected adults stated that they smoke regularly.
Suppose a die is rolled times and shows three on top 36 times, for a sample proportion of 0. Previous Section. Table of Contents.
Next Section. To understand the meaning of the formulas for the mean and standard deviation of the sample proportion. Find the probability that the sample proportion computed from a sample of size will be within 5 percentage points of the true population proportion. Thus P 0. Compute the sample proportion of items shipped within 12 hours.
Confirm that the sample is large enough to assume that the sample proportion is normally distributed. When the sample size is large the sample proportion is normally distributed. Compute the sample proportion.
Find the sample proportion. Since we have a two-tailed test , the P-value is the probability that the z-score is less than Note: If you use this approach on an exam, you may also want to mention why this approach is appropriate. Specifically, the approach is appropriate because the sampling method was simple random sampling, the sample included at least 10 successes and 10 failures, and the population size was at least 10 times the sample size.
Problem 2: One-Tailed Test Suppose the previous example is stated a little bit differently. Suppose the CEO claims that at least 80 percent of the company's 1,, customers are very satisfied.
Again, customers are surveyed using simple random sampling. The result: 73 percent are very satisfied. Based on these results, should we accept or reject the CEO's hypothesis?
Assume a significance level of 0.
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