Final answer:
The question involves performing a hypothesis test for a population standard deviation of calculator prices, using the Chi-Square distribution. One must calculate the Chi-Square test statistic and compare it to a critical value from a Chi-Square distribution table to determine if the claim that the standard deviation is greater than $15 is supported.
Step-by-step explanation:
The subject in question is a hypothesis test about a population standard deviation in the field of statistics, which is a branch of mathematics. The scenario described involves testing the claim about the standard deviation of calculator prices across various stores. Since this involves a sample standard deviation and the test of a population parameter, we will use the Chi-Square (χ²) distribution for this hypothesis test.
Steps for the Hypothesis Test
- State the null hypothesis H0: σ² = 225 (where σ is the population standard deviation and the value 225 comes from 15², the claim squared).
- State the alternative hypothesis Ha: σ² > 225.
- Calculate the test statistic using the formula χ² = (n-1)s² / σ₀², where n is the sample size, s is the sample standard deviation, and σ₀ is the claimed standard deviation under the null hypothesis.
- Find the critical value from the χ² distribution table using the degrees of freedom (n-1) and the level of significance (usually α=0.05).
- Compare the test statistic to the critical value. If the test statistic is greater than the critical value, reject the null hypothesis.
For this test, using the formula for the test statistic:
χ² = (43-1)×12² / 15²
= 42×144 / 225
= 42×0.64
= 26.88
To determine if this test statistic is significant, compare it with the critical value for χ² with 42 degrees of freedom at the chosen significance level. If 26.88 exceeds the critical value, we reject the null hypothesis and conclude that there is sufficient evidence to support the claim that the standard deviation is greater than $15.