Chapter 6 – Estimation Procedures

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Questions and Answers

What is the value of Z for a 99% confidence interval?

3.29

2.58 (correct)

1.64

1.96

What is the formula to calculate the required sample size for a desired level of precision?

$n = Z \times \sigma + ME$

$n = \frac{Z^2 \times \sigma^2}{ME^2}$ (correct)

$n = \frac{Z \times \sigma}{ME}$

$n = \frac{ME^2}{Z^2 \times \sigma^2}$

How many older adults does the gerontologist need to sample to be 99% confident within a margin of error of $1,000?

200

167 (correct)

250

100

If the margin of error (ME) was increased to $1,500, what would be the effect on the required sample size?

Sample size would decrease Signup and view all the answers

What does the term σ represent in the sample size formula?

Population standard deviation Signup and view all the answers

What does efficiency in the context of an estimator primarily relate to?

The clustering of the sampling distribution about its mean Signup and view all the answers

How is the standard error of the mean calculated?

Population standard deviation divided by the square root of sample size Signup and view all the answers

Which sample size would yield a lower standard deviation of the sampling distribution?

Sample size of 1,000 Signup and view all the answers

If the population standard deviation is $5,000 and the sample size is 100, what is the standard deviation of the sampling distribution?

$500 Signup and view all the answers

What effect does increasing the sample size have on the efficiency of the estimator?

It increases clustering Signup and view all the answers

What is the standard deviation of all possible sample means for a sample of size 1,000 with a population standard deviation of $5,000?

$158.13 Signup and view all the answers

In terms of bias, how do both samples (n=100 and n=1,000) compare when sampling the same population?

Both samples are unbiased estimators Signup and view all the answers

What is the standard deviation of the sampling distribution for a sample size of 100 when the population standard deviation is $5,000?

$500.00 Signup and view all the answers

What is the sample proportion of respondents who are 'very happy' in the United States?

0.360 Signup and view all the answers

Which country has the highest number of respondents reporting they are 'very happy'?

United States Signup and view all the answers

At the 95% confidence level, what formula is typically used to estimate a confidence interval for a sample proportion?

p ± Z * sqrt((p*(1-p))/n) Signup and view all the answers

If 35 out of 150 respondents are 'extremely' interested in clubs, what is the estimated sample proportion?

0.23 Signup and view all the answers

What is the critical Z-value used when constructing a 95% confidence interval?

1.96 Signup and view all the answers

What is the average (X̄) occupational prestige score reported in the survey?

43.87 Signup and view all the answers

Which value represents the standard deviation for the respondents' average daily television watching?

2.20 Signup and view all the answers

What is the correct interpretation of a confidence interval?

It provides a range of values likely containing the population parameter. Signup and view all the answers

What does the mean number of cigarettes smoked per day for daily smokers in Canada represent?

An estimate of cigarette consumption for daily smokers. Signup and view all the answers

What is the calculated 95% confidence interval for the mean number of cigarettes smoked per day?

Between 13.76 and 16.19. Signup and view all the answers

How can SPSS be used to calculate sample proportions?

By using the Frequencies command to determine valid frequencies. Signup and view all the answers

What percentage of respondents reported having a seasonal flu shot in their lifetime?

60.5% Signup and view all the answers

What is necessary to compute the confidence interval for a sample proportion?

Sample statistics and sample proportions. Signup and view all the answers

Which command in SPSS is NOT relevant for calculating sample proportions?

Correlation Signup and view all the answers

What proportion corresponds to a percentage of 39.5%?

0.3950 Signup and view all the answers

What does Pu in the confidence interval formula represent?

The sample proportion for the studied variable. Signup and view all the answers

What key methodological information is required in published election period poll results?

The date on which the poll was conducted Signup and view all the answers

What is one significant benefit of the growth in opinion polling for social sciences?

Greater availability of nationally representative databases Signup and view all the answers

How does the General Social Survey (GSS) enhance research credibility?

By offering high-quality data for testing theories Signup and view all the answers

Since when has Statistics Canada conducted the General Social Survey (GSS)?

1985 Signup and view all the answers

What type of design is used to select the GSS sample?

Probability design based on EPSEM Signup and view all the answers

What does the GSS enable researchers to do regarding public opinion?

Assess trends in public opinion over time Signup and view all the answers

Which type of record does the GSS provide due to its longitudinal nature?

Longitudinal record of sentiment and opinion Signup and view all the answers

What is a characteristic of the sample size in the GSS?

It is in the thousands providing high precision Signup and view all the answers

What percentage of the sample agreed that marijuana should be legalized?

375 Signup and view all the answers

How many respondents attended a religious service at least once in the past month?

1023 Signup and view all the answers

What is the sample size used to construct confidence intervals for the population mean with a sample mean of 100 and standard deviation of 100?

500 Signup and view all the answers

What effect does increasing the confidence level have on the width of the confidence interval?

It increases the width. Signup and view all the answers

What was the sample proportion used throughout for the three sample sizes in constructing the confidence interval?

0.40 Signup and view all the answers

At what confidence level is the prediction about candidate A's margin of victory being assessed?

99% Signup and view all the answers

What is the required percentage of the vote for a candidate to be considered a winner in a two-candidate race?

51% Signup and view all the answers

What was the sample size used in the survey that found 51% of respondents preferred candidate A for mayor?

578 Signup and view all the answers

Study Notes

Estimation Procedures

Estimation procedures in inferential statistics aim to estimate population values from sample statistics.
Point estimates use a single sample statistic to estimate a population value.
Confidence intervals give a range of values (an interval) instead of a single point, providing a more comprehensive estimate.

Bias and Efficiency

Estimators are selected based on bias and efficiency.
An unbiased estimator has a sampling distribution mean equal to the population value.
Sample means and proportions are typically unbiased.
Efficiency refers to clustering of the sampling distribution around its mean; a more clustered distribution indicates higher efficiency.
Smaller standard deviations of sampling distributions correspond to higher efficiency.
Larger sample sizes lead to more efficient estimates.

Estimation Procedures: Introduction

Constructing a point estimate involves drawing a sample, calculating either a proportion or mean, and using that statistic to estimate the population parameter.
Larger samples produce more accurate point estimates.
Interval estimates, unlike point estimates, provide a range of values, rather than a single value.

Interval Estimation Procedures for Sample Means (σ Known)

A confidence interval based on sample means utilizes a formula that accounts for the known population standard deviation.
The formula accounts for the sample mean and a relevant Z-score.

Interval Estimation Procedures for Sample Means (σ Unknown)

Sample standard deviation is used as a substitute for the population standard deviation when it is unknown.
The formula adapted to the student's t distribution.
Sample size, greater than 100, can be assumed for a large sample.

Interval Estimation Procedures for Sample Proportions (Large Samples)

Estimation procedures for sample proportions are similar to those for sample means.
The major difference is the use of a different sampling distribution due to the different statistic.
The Central Limit Theorem allows for the assumption of a normal distribution when nP and n(1-P) are 15 or larger.

Graphing a Confidence Interval of a Sample Mean

Confidence intervals for sample means are graphically depicted using error bars, where the error bar spans the confidence interval.

Determining Sample Size

Sample size needed to attain a desired confidence interval is determined before conducting a study.
Rearranging the formulas can be used to determine the sample size needed for various confidence interval estimates.

Interpreting Statistics: Predicting the Election of the Government of Canada

Statistical methods for estimating population parameters are used in many ways.
Statistical techniques are applied to predict elections and estimate political sentiment.
Confidence intervals are used to estimate population values which provide a range of possible values, often expressed as percentages.

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Test your knowledge on sample size calculations for a 99% confidence interval. This quiz covers key concepts such as the Z value, margin of error, and the significance of σ in sample size formulas. Perfect for students studying statistics or research methodology.