# Use R 1. A light bulb manufacturer claims their light bulbs as for 1,000 hours. You...

###### Question:

Use R
1. A light bulb manufacturer claims their light bulbs as for 1,000 hours. You sample 50 light bulbs and you find that the sample average light bulb lifetime duration is 955 hours. Assume that you know the population standard deviation, and that it is 220 hours. That is n 50、 x"= 955, σ = 220 Test the null hypothesis Ho: μ-1,000 against the alternative hypothesis Ha: μ < 1,000 at the 0.05 significance level a. Calculate the critical statistic using the qnorm function. Compare this to the test statistic, found using the sample information. Do you reject, or fail to reject the nul hypothesis? Find the p-value associated with the sample statistic under the null hypothesis. You can do this using the pnorm function. Report the p-value. Is your conclusion consistent with part (a)? b. 2. Consider the scenario presented in question 1. Here we will consider how the p-value will change with the sample size, keeping all the other information the same Create a vector, called "n", of different sample sizes, from 10 to 200 at an increment of 5 You can do this using the seq) command Use this vector of sample sizes, along with the information in question1 and the pnormO function, to create a vector of p-values that is the same length as the vector n Plot the vector of sample sizes on the horizontal axis and the vector of p-values on the vertical axis using the plot function Label the axes. Paste the figure below. Wha What range of sample sizes will give you a statistically significant result, holding the population standard deviation and sample average constant? a. t happens to the p-value as the sample size increases? b.
3. Now we will see how the sample average impacts the p-value Create a vector, called "xbar", of different observe increment of 5 Use this vector of sample averages, along with the information in question 1 and the pnorm0 function, to create a vector of p-values that is the same length as "xbar" Plot the vector of sample averages on the horizontal axis and the vector of p-values on the vertical axis using the plot function. Label the axes. d sample averages, from 920 to 1000 at an Paste the figure below. Wh What range of sample averages will give you a statistically significant result, holding the population standard deviation and sample size constant? a. at happens to the p-value as the sample average increases? b. 4. Now, we'll go through the same procedure keeping sample size and sample mean constant but changing the population standard deviation Create a numerical vector, called " to 250 at an increment of 5. You can do this using the seq0 command. Use this vector of population standard deviations, along with the information in question 1 and the pnorm function, to create a vector of p-values. Plot the vector of population standard deviations on the horizontal axis and the vector of p-values on the vertical axis using the plot function. Label the axes sigma", of different population standard deviations, from 120 Paste the figure below. What happens to the p-value as the population standard deviation increases? What range of population standard deviations will give you a statistically significant result, holding the sample average and sample size constant? a. b.
S. Now we'"ll do power analysis for question I For the hypothesis Ho 1,000 vs. Ha: μ < 1,000, with n 50, χ"z 955, σ 220, with α = 0.05, there is a range of values of x that will result in us rejecting the null hypothesis. This is called the rejection region, and the figure you created in question three hints at this. Find the rejection region in terms of x- Now consider alternative true values of μ ranging from 800 to 1000 at intervals of ten. Create a vector of these values use the seq0 function, and call this vector "true mu" a. b. c. Use the pnorm) to calculate the probability of observing the critical sample average found in part a., or a smaller value of x, using the vector of true mean values. This is the power of the test for alternative true values of mu. Plot the vector of alternative true values of μ on the horizontal axis and the vector a power on the vertical axis. Paste the figure below For what range of true values offl does our test have a power of 80% or more? d. e.

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