t-Test

Problem :

A coffee shop owner claims that the average caffeine content in their new espresso blend is 120 mg per serving. A random sample of 30 servings was tested, revealing an average caffeine content of 125 mg with a standard deviation of 10 mg. Determine whether there is significant evidence to suggest the true caffeine content is different from the owner's claim.

Step 1: State the Hypothesis

Null Hypothesis (Ho): μ = 120 (The average caffeine content is 120 mg)

Alternative Hypothesis (Ha): μ ≠ 120 (The average caffeine content is different from 120 mg)

Step 2: Identify Given Values

Significance level: α = 0.05 (95% confidence level)

Sample size: n = 30

Sample mean: x̄ = 125

Claimed population mean: μ = 120

Standard deviation: s = 10

Step 3: Compute the t-Statistic (use t-test calculator for verification)

Using the t-distribution formula:

Step 4: Determine Degrees of Freedom (df)

𝑑𝑓 =  𝑛 −  1

     = 30 − 1 = 29

Step 5: Find the Critical t-Value

Using a t-table or calculator for df = 29 and α = 0.05 (two-tailed test), the critical t-value is approximately ±2.045.

Step 6: Compare and Make a Decision

Since the calculated t-value (2.74) is greater than the critical value (2.045), we reject the null hypothesis.

Conclusion:

There is significant evidence to suggest that the actual caffeine content in the espresso blend is different from the claimed 120 mg.

Can You Do These?

Directions: Solve the following problems and check your answers using the provided t-test calculator.

Problem 1: Fast Food Calories 🍔

A restaurant claims that their grilled chicken sandwich contains an average of 500 calories. A random sample of 36 sandwiches was tested, revealing an average calorie content of 510 calories with a standard deviation of 20 calories. Test if the true calorie count is different from the claimed 500 calories at α = 0.05.

Problem 2: Video Game Reaction Time 🎮

A game developer claims that the average reaction time of players in a new FPS game is 350 milliseconds. A test with 40 players showed an average reaction time of 342 milliseconds with a standard deviation of 15 milliseconds. Test if the average reaction time is significantly different from 350 ms at α = 0.05.

Problem 3: Student Study Hours 📚

A university states that students study for an average of 15 hours per week. A survey of 25 students found an average of 13 hours with a standard deviation of 5 hours. Test if the study hours are significantly different from the claim at α = 0.05.

Problem 4: Energy Drink Caffeine Content ☕

An energy drink company claims that each can contains 160 mg of caffeine. A lab test of 30 cans showed an average caffeine content of 165 mg with a standard deviation of 8 mg. Test if the actual caffeine content is different at α = 0.05.

Problem 5: Roller Coaster Enjoyment 🎢

An amusement park claims that their new roller coaster has an average enjoyment score of 90. A group of 35 visitors who rode the coaster gave it an average score of 87, with a standard deviation of 8.  At α = 0.05, is there enough evidence to say the actual enjoyment score is different from 90?

Solo osol

By: Chito Rabago