Confidence Intervals for a Single Population Mean
Answering “What is the population mean?”
�Removing an Unreasonable Assumption
�The t-distributions
The tails of the t-distributions are “fatter” (higher up) than the normal distribution
The t-distributions have more area (probability) in the tails of the distribution
As the degrees of freedom increase, the t-distribution gets closer to the normal distribution
�Calculating Probability Under a t-distribution
�Calculating Probability Under a t-distribution
�Calculating Critical Values on a t-distribution
Because the t-distributions differ from the standard normal distribution, we’ll need to calculate the critical value we use when constructing confidence intervals
Example: Find the critical value for a 90% confidence interval, using a t-distribution with 11 degrees of freedom
�Calculating Critical Values on a t-distribution
Because the t-distributions differ from the standard normal distribution, we’ll need to calculate the critical value we use when constructing confidence intervals
Example: Find the critical value for a 90% confidence interval, using a t-distribution with 11 degrees of freedom
�Calculating Critical Values on a t-distribution
Because the t-distributions differ from the standard normal distribution, we’ll need to calculate the critical value we use when constructing confidence intervals
Example: Find the critical value for a 90% confidence interval, using a t-distribution with 11 degrees of freedom
�Calculating Critical Values on a t-distribution
Because the t-distributions differ from the standard normal distribution, we’ll need to calculate the critical value we use when constructing confidence intervals
Example: Find the critical value for a 90% confidence interval, using a t-distribution with 11 degrees of freedom
�Calculating Critical Values on a t-distribution
Because the t-distributions differ from the standard normal distribution, we’ll need to calculate the critical value we use when constructing confidence intervals
Example: Find the critical value for a 90% confidence interval, using a t-distribution with 11 degrees of freedom
�Calculating Critical Values on a t-distribution
Because the t-distributions differ from the standard normal distribution, we’ll need to calculate the critical value we use when constructing confidence intervals
Example: Find the critical value for a 90% confidence interval, using a t-distribution with 11 degrees of freedom
�Summary (So Far…)
�Examples
�Examples: Video Game Playtime
Scenario: A gaming company is analyzing player engagement for their newest game. They collected data from 42 players, finding an average daily playtime of 3.6 hours with a standard deviation of 0.8 hours. Use the sample data to construct a 95% confidence interval for the mean daily playtime of all players.
Solution. We need to evaluate the expression for a confidence interval
(𝚙𝚘𝚒𝚗𝚝 𝚎𝚜𝚝𝚒𝚖𝚊𝚝𝚎) ± (𝚌𝚛𝚒𝚝𝚒𝚌𝚊𝚕 𝚟𝚊𝚕𝚞𝚎)⋅(𝚜𝚝𝚊𝚗𝚍𝚊𝚛𝚍 𝚎𝚛𝚛𝚘𝚛)
Finding the Critical Value:
�Examples: Video Game Playtime
Scenario: A gaming company is analyzing player engagement for their newest game. They collected data from 42 players, finding an average daily playtime of 3.6 hours with a standard deviation of 0.8 hours. Use the sample data to construct a 95% confidence interval for the mean daily playtime of all players.
Solution. We need to evaluate the expression for a confidence interval
(𝚙𝚘𝚒𝚗𝚝 𝚎𝚜𝚝𝚒𝚖𝚊𝚝𝚎) ± (𝚌𝚛𝚒𝚝𝚒𝚌𝚊𝚕 𝚟𝚊𝚕𝚞𝚎)⋅(𝚜𝚝𝚊𝚗𝚍𝚊𝚛𝚍 𝚎𝚛𝚛𝚘𝚛)
Finding the Critical Value:
�Examples: Video Game Playtime
Scenario: A gaming company is analyzing player engagement for their newest game. They collected data from 42 players, finding an average daily playtime of 3.6 hours with a standard deviation of 0.8 hours. Use the sample data to construct a 95% confidence interval for the mean daily playtime of all players.
Solution. We need to evaluate the expression for a confidence interval
(𝚙𝚘𝚒𝚗𝚝 𝚎𝚜𝚝𝚒𝚖𝚊𝚝𝚎) ± (𝚌𝚛𝚒𝚝𝚒𝚌𝚊𝚕 𝚟𝚊𝚕𝚞𝚎)⋅(𝚜𝚝𝚊𝚗𝚍𝚊𝚛𝚍 𝚎𝚛𝚛𝚘𝚛)
Finding the Critical Value:
�Examples: Video Game Playtime
Interpretation: We are 95% confident that the true average daily play time for players of this new game is between 3.35 hours (3 hours, 21 minutes) and 3.85 hours (3 hours, 51 minutes).
�Examples: Coffee Shop Spending
Scenario: A coffee shop is interested in estimating the average amount customers spend during a visit. A sample of 30 customers revealed an average spending of $8.45 with a standard deviation of $2.12. Construct a 90% confidence interval for the mean amount spent by customers at this coffee shop.
�Examples: Daily Text Messages Sent
Scenario: A phone company is analyzing the texting habits of teenagers. From a sample of 25 teenagers, the average number of text messages sent daily is 34.3, with a standard deviation of 12.7 messages. Determine a 95% confidence interval for the average daily number of text messages sent by teenagers.
�Examples: Streaming Service Viewing Time
Scenario: A streaming service wants to estimate the weekly viewing habits of its users. Data from 60 users show an average of 12.4 hours spent watching shows or movies each week, with a standard deviation of 4.2 hours. Calculate a 90% confidence interval for the mean weekly viewing time of all users.
�Examples: Concert Ticket Prices
Scenario: A group of regular concert-goers is interested in estimating the average ticket price for popular concerts in their city. A sample of 20 concerts shows an average ticket price of $92.15, with a standard deviation of $13.58. Determine a 95% confidence interval for the mean ticket price.
�Examples: Sleep Duration for Young Adults
Scenario: A health organization is examining the sleep habits of young adults aged 18–24. Data from 45 individuals show an average nightly sleep duration of 6.9 hours with a standard deviation of 1.2 hours. What is a 98% confidence interval for the mean nightly sleep duration of young adults?
�Examples: Delivery Times for Food Orders
Scenario: A food delivery app wants to estimate the average delivery time for its orders. A sample of 38 orders shows an average delivery time of 27.4 minutes, with a standard deviation of 5.9 minutes. Calculate a 99% confidence interval for the mean delivery time of all orders.
Example: Monthly Social Media Posts by Influencers
Scenario: A social media marketing agency is studying the posting habits of influencers. A sample of 28 influencers shows an average of 49.2 posts per month, with a standard deviation of 7.3 posts. Determine a 95% confidence interval for the mean number of posts made by influencers each month.
�Summary
�Next Time…