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 (σ2\sigma^2) of a normal population, the appropriate test is:

  • One variance χ2\chi^2 Hypothesis Test.

Assumptions:

The following assumptions apply:

  1. Population standard deviation is unknown.
  2. Simple random sample (data was collected randomly over 16 days).
  3. 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 (σ\sigma) in volts. The hypotheses are:

Null Hypothesis (H0H_0):

H0:σ=0.21or equivalently, σ2=0.212=0.0441.H_0: \sigma = 0.21 \quad \text{or equivalently, } \sigma^2 = 0.21^2 = 0.0441.

Alternative Hypothesis (HaH_a):

Ha:σ<0.21or equivalently, σ2<0.0441.H_a: \sigma < 0.21 \quad \text{or equivalently, } \sigma^2 < 0.0441.

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 χ2\chi^2-test is: χ2=(n1)s2σ02,\chi^2 = \frac{(n-1)s^2}{\sigma_0^2}, where:

  • n=16n = 16 (sample size),
  • s=0.1s = 0.1 (sample standard deviation),
  • σ0=0.21\sigma_0 = 0.21 (hypothesized population standard deviation).
  1. Compute s2s^2 (sample variance): s2=(0.1)2=0.01.s^2 = (0.1)^2 = 0.01.

  2. Compute the test statistic: χ2=(161)(0.01)0.0441=15×0.010.0441=0.150.04413.40.\chi^2 = \frac{(16-1)(0.01)}{0.0441} = \frac{15 \times 0.01}{0.0441} = \frac{0.15}{0.0441} \approx 3.40.


Step 4: Critical Value and Decision Rule

The critical value for a left-tailed χ2\chi^2-test at α=0.01\alpha = 0.01 with df=n1=15df = n-1 = 15 can be obtained from the χ2\chi^2-distribution table or using statistical software.

χ0.01,152=6.262.\chi^2_{0.01, 15} = 6.262.

Decision Rule:

  • Reject H0H_0 if χ2<6.262\chi^2 < 6.262.

Step 5: Conclusion

The test statistic χ2=3.40\chi^2 = 3.40 is less than the critical value 6.2626.262. 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:

  1. Would you like to see the χ2\chi^2-distribution graph for this problem?
  2. Do you need help understanding why the χ2\chi^2-test was chosen?
  3. Should we verify this result using software or simulation?
  4. Would you like an explanation of the assumptions in detail?
  5. 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)