FINAL PROJECT Interval Estimate Proportion of shoppers using resusable bags






Shoppers






Do you use resusable bags / recyclable bags?






Name






Professor Silberman






MAT 177: INTERVAL ESTIMATE PROJECT






Date

Question: What is the proportion of people in your area who use “reusable" bags when  grocery shopping instead of the usual plastic or paper bags provided by the grocery stores? 0.56

 The sample consist of 50 respondents of shoppers in a supermarket 28 affirmed that they use the reusable bags when grocery shopping and 22 stated they do not.

Collect a sample and use the data from the sample to estimate the proportion of all shoppers in your area who use reusable bags. Write a clear and concise report. Some decisions to make: Sample size? Sampling method? Which grocery stores? Best observation point: from the car or standing outside the store?  What days and times of day? Etc…

 

Report requirements: Type your name on top of the first page

a. At least two pages long, single spaced and with Arial font size 12. b. Check grammar, sentence structure and spelling. (Points will be deducted).

c. The calculator screens are same size as the ones shown and are incorporated in the text. d. Page Layout: 0.9” margins all around for calculator screens to fit properly.

NO LATE PROJECT ACCEPTED. LATE= ZERO

e.  Project must be typed (Points will be deducted)

f. Minimum confidence level you may use is 85%; Maximum error estimate is 8%.

g. There are 2 pages of questions to be discussed.

h. You must review this project at the writing center and make suggested changes; this is not optional.

 

Minimum Content:

1. Explain the difference between point estimate and interval estimate and explain the advantages of one over the other.

The point estimate is calculated from the information obtained from the sample and that is used to estimate the population parameter population parameter (such as the mean (μ), or the standard deviation (σ)). The interval estimate establishes the range of values ​​where the parameter is most likely to be found. This is based on the probability of occurrence, value of the population parameter and the confidence interval. In the point estimate approach one sample statistic is used as the point estimate of the population parameter.   The point estimate approach is also straightforward to use compared to the interval estimate. The interval estimate of a parameter is useful as it shows the range of values within which the parameter has a specified probability lying within that range. Increasing the sample size ensures there is better precision and narrows the confidence intervals.

1. What population could you reasonably approximate based upon your sample? The world? The United States?.........Explain your answer.

The population of people using the reusable bags in the city and the United States could be estimated using as this represents what is to be expected. People in different areas have diverse shopping habits and those in cities frequent the supermarkets more

 

3.       Explain and justify your choice of sample size. Describe in detail how you obtained your data. State the type of sample you obtained (see chapter 1). Data must be included. Use the formula for the margin of error to explain why this happens

 50 people were randomly selected to respond to the question and the simple random sampling method. The shoppers who responded gave a yes or no answer on whether they used the reusable bags 28 stated that they did, and assuming that the respondents would represent what would be expected in the population. This information was used in the data analysis and parameter estimate. The margin of error is calculated as the product of the critical value (1.44) for 85% CI and the standard deviation.

4.    Does your project satisfy the statistical requirements needed for computing interval estimates of a proportion? (see section 7.1)

The project satisfies the statistical requirements for interval estimates of the proportion as there is a simple random sample of size 50, and the samples are chosen independently from each other. The answer ‘yes’ represents success and this is used in the sample proportion as the total frequency of the ‘yeses’ divided by the sample size is the sample proportion (p̂). There is also a confidence level provided (confidence level Z=1.44 for 85% confidence).

5. List the possible bias or shortcomings of your sampling method.

• In the simple random sample, people are chosen by chance, and the sample may not fully represent the population.


• There is likely to be response bias if the  people sampled believe that certain answers make them look better and answer what they think Is expected.   


• The approach may also require large sample size so that the sample can be applied appropriately.

6. Use the calculator to find the interval estimate. Express it in two different ways: interval form as indicated in the calculator screen below and in the form p^ˆ ± E .

 






Upper boundàp̂+(Z+ std dev.)=0.56+(1.44* 0.0702)= 0.6611      






Lower boundàp̂+(Z- std dev.)-0 .56-(1.44* 0.0702) = 0.4589






85% Confidence Interval: 0.56 ± 0.1011 (0.4589 to 0.6611). With 85% confidence the population mean is between 0.4589 and 0.661, based on 50 samples






 






 

7. What is the definition of the margin of error? What is the formula for finding the margin of error?

 






 The margin of error is confidence interval for a particular statistic, which explains the results that are due to the sampling errors. Total population does not contain the sampling errors, and the margin of error depends on the confidence interval radius for the statistic.






The Margin of Error (MOE) is calculated according to the formula: MOE = z * √p * (1 - p) / √n

Where: z = 1.44 for a confidence level (α) of 85%, p = proportion (expressed as a decimal), n = sample size.

z = 1.44, p = 0.56, n = 50

MOE = 1.44 * √0.56 * (1 - 0.56) / √50

MOE = 0.715 / 7.071 * 100 = 10.109%

The margin of error is ±10.109%

8. Write about the effect of sample size and confidence level on the width of the interval estimate. Does a larger sample make the interval wider or narrower? Does a higher confidence level make the interval wider or narrower? (See A) (INCREASE N, DECREASE e>>>DECrease interval)  Inc. Confidence Level , increase E>>>incr. Confidence Interval

 There is a tradeoff between precision and the width of the interval, and when the sample size is larger the interval becomes more precise and therefore less extensive. More information (sample) increases more precision in the estimate and the width of the interval estimate gets narrower (Triola 117).  Increasing the confidence level (percent confidence) requires increasing the range of values and hence increasing the width of the interval. Increasing the confidence in an estimate is linked to increasing the width of the interval.

9. Explain the meaning of confidence level (pay attention to the definition in the text). What does it mean to say: we are 95% confident that the population proportion is in a certain interval?

 






The confidence level is the probability that the parameter to be estimated is in the confidence interval, and the confidence level (p) is represented as (1- α), and the level of confidence is α. The 95% confidence interval for the population proportion, means that there is there is a 95% probability that the population proportion lies between a certain interval






10. Show three calculator screens for the proportion estimates and integrate them in  your report.






Upper boundàp̂+(Z+ std dev.)=0.56+(1.44* 0.0702)= 0.6611      






Lower boundàp̂+(Z- std dev.)-0 .56-(1.44* 0.0702) = 0.4589

 

11. Evaluate the importance of your results and state who may be interested in them.

   The results are important to calculate the proportion estimates and this requires sample size and population parameters. This is useful to the supermarket, the reusable bag manufacturers, and local governments that have implemented policies on the adoption of these bags.

12) Discuss the following: If you were writing for a Master’s Thesis, and would receive a $10,000 cash payment if your proposal were accepted, what changes would you make to improve the reliability and accuracy of this project? What would you do differently in order to get a more valid result? (This discussion should not be limited to discussing the topics in Chapter 7).

 There is a need to reduce the variability, which affects the reliability of the research w here there is emphasis on measuring the required attribute(s) or dimensions.  When this is prioritized it ensures that the measures are consistent and results dependable (Triola 99). Collecting data on the same phenomenon in multiple sites increases the validity of the results, where the inferences made then are more accurate. Furthermore, if there are changes on the dependent variable not directly linked to the independent variable (s) then there is a need for further investigations.

13) In general (not specific to this project), when would you accept a lower confidence level? Give an example of a population you might sample and accept a lower confidence level. When would you require a higher confidence level? Give an example of a population you might sample and require a higher confidence level.

 






As the sample size increases the confidence interval gets smaller. For instance, when taking census to determine certain demographic profiles the samples are mostly large. An example of  population requiring a higher confidence interval is a sample of political preference to increase accuracy some people may hold positions across the political spectrum.

3A) Based on information given in this project, there is a minimum sample size that must be used. What is the minimum sample size? (show your work).

 Assuming the population proportion is 50%, the sample size would be






n= z² * p̂ (1- p̂)/ e2=1.44^2*(0.5*0.5)/8%^2=81

 

What might be an advantage to collecting a sample larger than the minimum sample size?

 A large sample size minimizes the margin of errors, and increasing the level of confidence requires increases the sample size. The sample should reflect the essential characteristics of the population to be a good representative and when increasing the sample size beyond the minimum sample size this needs to be considered.

Include the table you used for the for the grocery store assignment.(assuming you did it correctly; if not, fix it)You must show each of the stores you considered sampling in your town, the values you used in randint( ) function, and identify where you actually collected your sample(s). You need to show similar information for the sampling intervals. Remember you are probably sampling more than 1 grocery store, so you must use randint() function correctly; same requirements for sampling intervals.