A report states that the cost of repairing a hybrid vehicle is falling even while typical repairs on conventional vehicles are getting more expensive. The most common hybrid repair, replacing the hybrid inverter assembly, had a mean repair cost of $ 3 comma 927 3,927 in 2012. Industry experts suspect that the cost will continue to decrease given the increase in the number of technicians who have gained expertise on fixing gas-electric engines in recent months. Suppose a sample of 100 100 hybrid inverter assembly repairs completed in the last month was selected. The sample mean repair cost was $ 3 comma 850 3,850 with the sample standard deviation of $ 300 300. Complete parts (a) and(b) below. a. Is there evidence that the population mean cost is less than $ 3 comma 927 3,927? (Use a 0.10 0.10 level ofsignificance.) State the null and alternative hypotheses. H0: mu μ greater than or equals ≥ $ 3927 3927 H1: mu μ less than < $ 3927 3927 (Type integers or decimals.) Find the test statistic for this hypothesis test. t Subscript STAT tSTAT equals = nothing(Round to two decimal places as needed.) The critical value(s) for the test statistic is(are) nothing . (Round to two decimal places as needed. Use a comma to separate answers as needed.) Is there sufficient evidence to reject the null hypothesis using alpha α equals = 0.10 0.10? A. Do not reject Do not reject the null hypothesis. There is insufficient insufficient evidence at the 0.10 0.10 level of significance that the population mean cost is less than $ 3 comma 927 3,927. B. Reject Reject the null hypothesis. There is sufficient sufficient evidence at the 0.10 0.10 level of significance that the population mean cost is greater than $ 3 comma 927 3,927. C. Reject Reject the null hypothesis. There is sufficient sufficient evidence at the 0.10 0.10 level of significance that the population mean cost is less than $ 3 comma 927 3,927. D. Do not reject Do not reject the null hypothesis. There is insufficient insufficient evidence at the 0.10 0.10 level of significance that the population mean cost is greater than $ 3 comma 927 3,927. b. Determine the p-value and interpret its meaning.
Accepted Solution
A:
Answer:a. Ab. p-value= 0.5Step-by-step explanation:Hello!First, you need to deterHello!You are studying the cost of replace of the hybrid inverter of hybrid vehicles, this is your study variable, which has a known mean (μ) of $3.927. For this a sample of 100 hybrid inverter assembly repairs completed in the last month. The sample mean x[bar] is $3.850 and the sample standard deviation is $300a. You need to test the hypothesis that "the population mean of the cost of replacement of the hybrid inverter is less than $3.927"Your statistical hypothesis is:H₀:μ≥3.927H₁:μ<3.927Level of significance:α: 0.10For this problem, considering we have a sample large enough, I'll use the Central Limit Theorem and approximate the sample mean distribution to normal. That way I can use the statistic Z to resolve the test.Z= (x[bar]-μ)/(σ/√n)≈N(0;1)This test is one-tailed to the left, this means we will work with only one critical value to determine the rejection region.[tex]Z_{\alpha }[/tex] ⇒ [tex]Z_{0.10}[/tex]= -1.28So we will reject the null hypothesis if the calculated Z value is less or equal to -1.28 and support it if its greater than -1.28.Z= (x[bar]-μ)/(σ/√n) ⇒ Z= (3.850-3.927)/(300/10)= -0.00256≅ -0.003With this value, we can say, that at a level of significance of 0.10 there is insufficient evidence to reject the null hypothesis, in other words, the population mean cost is greater than $3.927.b.P-value is the probability of obtaining the results as extreme as the calculated results, under the assumption that the null hypothesis is correct. To decide for a statistical test using the p-value we need to compare it with the level of significance of the test. If the p-value > α you do not reject the null hypothesis.If the p-value < α you reject the null hypothesis.In this problem: P(Z≤-0.003)= 0.5p-value= 0.5 since it's greater than the significance level 0.10, we do not reject the null hypothesis.I hope you have a Super day!