Using Confidence Intervals to Test Hypotheses When analyzing the last digits of telephone numbers in Port Jefferson, it is found that among 1000 randomly selected digits, 119 are zeros. If the digits are randomly selected, the proportion of zeros should be 0.1.


c. Use the sample data to construct a 95% confidence interval estimate of the proportion of zeros. What does the confidence interval suggest about the claim that the proportion of zeros equals 0.1?

Verified step by step guidance

1. Step 1: Identify the problem and the given data. The problem involves constructing a 95% confidence interval for the proportion of zeros in a sample. The given data includes the sample size (n = 1000), the number of zeros (x = 119), and the hypothesized proportion (p = 0.1).
2. Step 2: Calculate the sample proportion (p̂). The sample proportion is calculated as p̂ = x / n, where x is the number of zeros and n is the sample size. Substitute the values x = 119 and n = 1000 into the formula.
3. Step 3: Determine the standard error (SE) of the sample proportion. The formula for the standard error is SE = sqrt((p̂ * (1 - p̂)) / n). Substitute the calculated value of p̂ and the sample size n = 1000 into the formula.
4. Step 4: Find the critical value (z) for a 95% confidence level. For a 95% confidence interval, the critical value z is approximately 1.96. This value corresponds to the z-score that captures the middle 95% of the standard normal distribution.
5. Step 5: Construct the confidence interval. The formula for the confidence interval is p̂ ± z * SE. Substitute the values of p̂, z, and SE into the formula to calculate the lower and upper bounds of the confidence interval. Interpret the confidence interval to determine whether it supports or refutes the claim that the proportion of zeros equals 0.1.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence, typically 95%. It provides an estimate of uncertainty around a sample statistic, allowing researchers to infer about the population. For example, if a 95% confidence interval for a proportion is calculated, it means that if the same sampling method were repeated many times, approximately 95% of the intervals would contain the true proportion.

Proportion

In statistics, a proportion is a type of ratio that represents a part of a whole, often expressed as a fraction or percentage. In the context of the question, the proportion of zeros in the sample is calculated by dividing the number of zeros (119) by the total number of digits (1000), yielding a sample proportion. This value is crucial for constructing the confidence interval and testing hypotheses about the population proportion.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population based on sample data. It involves formulating a null hypothesis (e.g., the proportion of zeros equals 0.1) and an alternative hypothesis (e.g., the proportion of zeros does not equal 0.1). The results from the confidence interval can be used to determine if the null hypothesis can be rejected, providing insight into whether the sample data supports or contradicts the claim about the population proportion.