STAT 1043 Homework 2

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School

New Brunswick Community College, Fredericton *

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1043

Subject

Statistics

Date

Feb 20, 2024

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docx

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Uploaded by wingsNDGRILL

STAT 1043 Homework 2 – Confidence Intervals Please submit through Dropbox by 5:30 pm on Thursday, January 25, 2024 Goal: A few more practice problems relating to samples and confidence intervals Directions: Please label your solutions properly and write as clearly as you can. Please ensure that the solutions are written in order. Collaboration in solving problems on the assignments are acceptable; however, copying assignments is inappropriate and will be considered academic misconduct. Please round final answers to 3-4 decimal places unless otherwise specified. For the calculation questions, it’s fine to use Excel or other software, but please take a picture/screenshot of your setup with the final answer or explain what you did on the software in writing to get your result. Activity: (The part to hand in for marks) 1. (2 marks) In your own words, explain the need for obtaining and using samples. Obtaining and using samples is important because it helps us understand and analyze larger things by studying smaller parts of them. Imagine you have a big bag of assorted candies. Instead of eating the entire bag at once, you might take small samples to get an idea of the different flavors. Similarly, in various fields like science, research, and statistics, taking samples allows us to draw conclusions about a whole population or system without having to study every single part of it. Samples provide a manageable way to gather information, make predictions, and understand the bigger picture. 2. (5 marks) In a sample of 50 days taken at Piece of Cake Bakery, it was found that an average of 75 cakes per day were sold. From similar past surveys, the population standard deviation was found to be 25 cakes per day. a. Determine the standard error ( σ √ n ) of cakes sold per day. 3.953 fi. 95.fi b. Determine the 90% confidence interval for the population mean number of cakes sold per day. ( 68.50, 81.50) c. If you wanted a wider interval, would you increase or decrease the confidence level? increase 3. (3 marks) Find the following t-distribution values: a. 95% confidence, sample size 72. The value of t at 5% level of significance is 2.145 b. 80% confidence, sample size 29. The value of t at 2% level of significance is 2.50. c. 99% confidence, sample size 99. The value of t at 10% level of significance is 1.796 4. (6 marks) Gus Gale is the owner of Gale's Gas Garage. Gus would like to estimate the mean

number of liters (L) of gasoline sold to his customers. From his records, he selects a random sample of 60 sales and finds the mean number of liters sold is 40 with a population standard deviation of 10 L. a. What is the point estimate of the population mean? The point estimate of the population mean (μ) is the sample mean (x).In this case, Gus Gale found that the mean number of liters sold in his random sample of 60 sales is 40 liters. Point Estimate (x = 40 liters) b. Develop a 99% confidence interval for the population mean. The critical value for a 99% confidence interval can be found using a Z-table or a calculator. For a 99% confidence interval, the critical value is approximately 2.576. Confidence Interval = 40± ( 2.576 × 10 60 ) Calculating this gives the confidence interval. Confidence Interval ≈( 37.15,42.85) c. What is the margin of error of this interval? d. Interpret the meaning of part (b). The 99% confidence interval for the population mean number of liters sold is from 37.15 liters to 42.85 liters. This means that if we were to take many random samples of 60 sales from Gale's Gas Garage and calculate a confidence interval for each sample mean, we would expect about 99% of those intervals to contain the true population mean. The margin of error is the range within which we are reasonably confident the true population mean falls, and in this case, it is approximately ±3.32 liters.

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