Math Problem Statement
The proper operation of typical home appliances requires voltage levels that do not vary much. A simple random sample of 16 voltage levels (in volts) was observed and measured at a house on 16 different days:
119.9
120.3
120.1
120.1
120
119.8
120.3
120.1
119.9
119.9
120.1
120.2
120.2
120.1
120
120.1
The mean was 120.1 volts with a standard deviation of 0.1 volts. Under 1% significance level can you conclude that the population standard deviation of voltage levels at the house is less than 0.21? (Note: If the normality plot is not provided then assume that the population is normal.)
Procedure: Select an answer One variance χ² Hypothesis Test One mean T Hypothesis Test One mean Z Hypothesis Test One proportion Z Hypothesis Test
Assumptions: (select everything that applies)
Population standard deviation is unknown
Simple random sample
The number of positive and negative responses are both greater than 10
Sample size is greater than 30
Population standard deviation is known
Normal population
Part 2 of 5
Step 1. Hypotheses Set-Up:
H0: Select an answer σ² μ p =
, where ? σ μ p is the Select an answer population mean population standard deviation population proportion and the units are Select an answer amp volts joules ohm
Ha: Select an answer p μ σ² ? < ≠ >
, and the test is Select an answer Left-Tailed Two-Tailed Right-Tailed
Solution
We are conducting a hypothesis test to determine if the population standard deviation of voltage levels at the house is less than 0.21. Here’s the structured solution:
Step 1: Procedure Selection
Since the population standard deviation is unknown, and we are testing the variance () of a normal population, the appropriate test is:
- One variance Hypothesis Test.
Assumptions:
The following assumptions apply:
- Population standard deviation is unknown.
- Simple random sample (data was collected randomly over 16 days).
- Normal population (assumed normality since no plot is provided and the sample size is small).
Step 2: Hypotheses Set-Up
We are testing the population standard deviation () in volts. The hypotheses are:
Null Hypothesis ():
Alternative Hypothesis ():
This is a left-tailed test because the claim is that the standard deviation is less than 0.21.
Step 3: Test Statistic Calculation
The test statistic for a one variance -test is: where:
- (sample size),
- (sample standard deviation),
- (hypothesized population standard deviation).
-
Compute (sample variance):
-
Compute the test statistic:
Step 4: Critical Value and Decision Rule
The critical value for a left-tailed -test at with can be obtained from the -distribution table or using statistical software.
Decision Rule:
- Reject if .
Step 5: Conclusion
The test statistic is less than the critical value . Therefore, we reject the null hypothesis.
Final Statement:
At the 1% significance level, there is sufficient evidence to conclude that the population standard deviation of voltage levels at the house is less than 0.21 volts.
Would you like me to help interpret further or calculate using a different method?
Here are 5 follow-up questions:
- Would you like to see the -distribution graph for this problem?
- Do you need help understanding why the -test was chosen?
- Should we verify this result using software or simulation?
- Would you like an explanation of the assumptions in detail?
- Are there any related problems you'd like to solve?
Tip: Always verify the normality assumption when conducting variance tests, as deviations can affect the validity of the test.
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Math Problem Analysis
Mathematical Concepts
Hypothesis Testing
Chi-Square Test
Statistics
Variance and Standard Deviation
Formulas
Chi-square statistic: χ² = (n-1) * s² / σ₀²
Sample variance: s² = (sample standard deviation)²
Theorems
Chi-Square Distribution
Suitable Grade Level
Undergraduate Statistics or Advanced High School (Grades 11-12)
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