Inference for One Sample

In statistics, inference for one sample involves making conclusions about a population based on information from a single sample. The goal is to estimate a population parameter, such as a population mean or proportion, and to determine the level of uncertainty in that estimate.

To conduct inference for one sample, it is important to first define the population of interest and the parameter to be estimated. Then, a random sample is drawn from the population and descriptive statistics, such as the sample mean and standard deviation, are calculated. From these statistics, a confidence interval can be calculated or a hypothesis test can be performed.

One Sample Proportion

In statistics, a proportion is the fraction of a population that has a certain characteristic of interest. For example, the proportion of voters who support a particular candidate or the proportion of students who pass a certain exam.

Inference for one sample proportion involves making conclusions about the proportion of a population based on information from a single sample. The goal is to estimate the population proportion and determine the level of uncertainty in that estimate.

When discussion proportions, we sometimes refer to this as the Rule of Sample Proportions. According to the Rule of Sample Proportions, if np ≥ 10 and n(1-p) ≥ 10 then the sampling distributing will be approximately normal.

Rule of Sample Proportions

The rule of sample proportions, also known as the rule of large numbers, is a fundamental principle in statistics that describes the behavior of sample proportions as sample size increases.

The rule of sample proportions states that as the sample size increases, the sample proportion of individuals with a certain characteristic of interest will converge to the population proportion. In other words, the larger the sample size, the more accurate the estimate of the population proportion will be.

Mathematically, the rule of sample proportions can be expressed as:

p-hat → p as n → ∞

where p-hat is the sample proportion, p is the population proportion, and n is the sample size.

Conduct inference for one sample proportion

To conduct inference for one sample proportion, a random sample is drawn from the population of interest and the sample proportion is calculated as the number of individuals in the sample who have the characteristic of interest divided by the sample size. The sample proportion is then used to estimate the population proportion.

Confidence Intervals

Confidence intervals can be used to estimate the range of values that is likely to contain the true population proportion with a certain level of confidence.

The formula for calculating a confidence interval for one sample proportion is:

sample proportion ± (critical value x standard error)

where the critical value is determined based on the desired level of confidence and the standard error is calculated as:

sqrt((sample proportion x (1 – sample proportion)) / sample size)

Hypothesis Testing

Hypothesis testing can also be used to determine whether the sample proportion is significantly different from a hypothesized value. The null hypothesis typically assumes that the sample proportion is equal to the hypothesized value, while the alternative hypothesis assumes that the sample proportion is not equal to the hypothesized value.

The test statistic is calculated as:

(test statistic) = (sample proportion – hypothesized proportion) / standard error and is compared to a critical value based on the desired level of significance and degrees of freedom.

When conducting inference for one sample proportion, it is important to consider any assumptions about the population, such as independence and random sampling, and to check whether these assumptions are met by the data. If the assumptions are not met, alternative methods may need to be used or adjustments may need to be made to the analysis.

Formal symbols and equations for conducting inference for one sample proportion.

For the following procedures, the assumption is that both np ≥ 10 and n(1-p) ≥ 10. When we’re constructing confidence intervals p is typically unknown, in which case we use p-hat as an estimate of p.

Confidence Intervals

Sample Proportion: p-hat = x/n,  where x is the number of individuals in the sample who have the characteristic of interest, and n is the sample size.

Standard Error: SE = sqrt((p-hat * (1 – p-hat)) / n)

Confidence Interval: p-hat ± z* SE, where z is the critical value from the standard normal distribution based on the desired level of confidence.

Finding the z* Multiplier

The value of the z multiplier or critical value depends on the level of confidence. The multiplier for the confidence interval for a population proportion can be found using the standard normal distribution [i.e., z distribution, N(0,1)]. The most commonly used level of confidence is 95%. As shown on the probability distribution plot below, the multiplier associated with a 95% confidence interval is 1.960, often rounded to 2.

Confidence level and corresponding multiplier
Confidence Level	z * Multiplier
90%	1.645
95%	1.960, often rounded to 2
98%	2.327
99%	2.576

Hypothesis Test

Here we will be using the five step hypothesis testing procedure to compare the proportion in one random sample to a specified population proportion using the normal approximation method.

1. Check assumptions and write hypotheses

In order to use the normal approximation method, the assumption is that both np0 ≥ 10 and n(1-p0) ≥ 10. p0 is the population proportion in the null hypothesis.

Null Hypothesis: H0: p = p0, where p0 is the hypothesized proportion.

Alternative Hypothesis: Ha: p ≠ p0.

2. Calculate the test statistic

When using the normal approximation method we will be using a z test statistic. The z test statistic tells us how far our sample proportion is from the hypothesized population proportion in standard error units.

Test Statistic: z = (p-hat – p0) / SE.

3. Determine the p-value

Given that the null hypothesis is true, the p value is the probability that a randomly selected sample of n would have a sample proportion as different, or more different, than the one in our sample, in the direction of the alternative hypothesis.

P-Value: the probability of obtaining a test statistic as extreme or more extreme than the observed value, assuming the null hypothesis is true.

4. Make a decision

• If p ≤ α, reject the null hypothesis (there is evidence to support the alternative hypothesis).
• If p > α, fail to reject the null hypothesis (there is not enough evidence to support the alternative hypothesis).

If significance level α is not mentioned then consider α = 0.05

Rejection Region: If the p-value is less than the level of significance (alpha), then the null hypothesis is rejected. The rejection region is defined by the critical values from the standard normal distribution.

Acceptance Region: If the p-value is greater than the level of significance (alpha), then the null hypothesis is not rejected. The acceptance region is defined by the complement of the rejection region.

5. State a “real world” conclusion

Based on all the 4 steps above, we should write a sentence or two concerning our decision in relation to the original research question.