Question

1) One of your employees has suggested that your company develop a new product. You decide to take a random sample of your customers and ask whether or not there is interest in the new product. The response is on a 1 to 5 scale with 1 indicating "definitely would not purchase"; 2, "probably would not purchase"; 3, "not sure"; 4, "probably would purchase"; and 5, "definitely would purchase." For an initial analysis, you will record the responses 1, 2, and 3 as "No" and 4 and 5 as "Yes." What sample size would you use if you wanted the 95% margin of error to be 0.2 or less? (Round your answer up to the next whole number.)

Participants

2)

An automobile manufacturer would like to know what proportion of its customers are dissatisfied with the service received from their local dealer. The customer relations department will survey a random sample of customers and compute a 95% confidence interval for the proportion that are dissatisfied. From past studies, it believes that this proportion will be about 0.22. Find the sample size needed if the margin of error of the confidence interval is to be no more than 0.035. (Round your answer up to the next whole number.)

Customers

Answer 1

Sample Size Variables Based on Target Population

Before you can calculate a sample size, you need to determine a few things about the target population and the sample you need:

1. Population Size — How many total people fit your demographic? For instance, if you want to know about mothers living in the US, your population size would be the total number of mothers living in the US. Not all populations sizes need to be this large. Even if your population size is small, just know who fits into your demographics. Don’t worry if you are unsure about this exact number. It is common for the population to be unknown or approximated between two educated guesses.
2. Margin of Error (Confidence Interval) — No sample will be perfect, so you must decide how much error to allow. The confidence interval determines how much higher or lower than the population mean you are willing to let your sample mean fall. If you’ve ever seen a political poll on the news, you’ve seen a confidence interval. For example, it will look something like this: “68% of voters said yes to Proposition Z, with a margin of error of +/- 5%.”
3. Confidence Level — How confident do you want to be that the actual mean falls within your confidence interval? The most common confidence intervals are 90% confident, 95% confident, and 99% confident.
4. Standard of Deviation — How much variance do you expect in your responses? Since we haven’t actually administered our survey yet, the safe decision is to use .5 – this is the most forgiving number and ensures that your sample will be large enough.

Calculating Sample Size

Okay, now that we have these values defined, we can calculate our needed sample size. This can be done using an online sample size calculator or with paper and pencil.

Your confidence level corresponds to a Z-score. This is a constant value needed for this equation. Here are the z-scores for the most common confidence levels:

• 90% – Z Score = 1.645
• 95% – Z Score = 1.96
• 99% – Z Score = 2.576

If you choose a different confidence level, use this Z-score table* to find your score.

Next, plug in your Z-score, Standard of Deviation, and confidence interval into the sample size calculator or into this equation:**

Necessary Sample Size = (Z-score)2 * StdDev*(1-StdDev) / (margin of error)₂

((1.96)2 x .5(.5)) / (.2)2
(3.8416 x .25) / .4
.9604 / .4
2.54

3 respondents are needed

Now for second problem

same formula

Necessary Sample Size = (Z-score)2 * StdDev*(1-StdDev) / (margin of error)₂

((1.96)2 x .5(.5)) / (.035)2
(3.8416 x .25) / .001225
.9604 / .001225
784

784 respondents are needed