Question

b. Using technology construct a 99% confidence interval.

Researchers are interested in the effect of a certain nutrient on the growth rate of plant seedlings. Using a hydroponics grow procedure that utilized water containing the nutrient, they planted six tomato plants and recorded the heights of each plant 14 days after germination. T heights, measured in millimeters, were 55.8, 59.6, 60.8, 63.6, 64.7, and 67.2 2. Complete parts a through d below. a. Using technology, find the 95% confidence interval for the population mean u. (OD) (Round to one decimal place as needed.)

Answer 1

Here in this scenario we have to calculate the Confidence Interval for population mean based on given sample of six tomato plants and the recorded Height in mm has mean 61.95 and sample Standerd deviation is 4.0624.

Now here we have to use t distribution because the population standard deviations is unknown.

Count, N: 6 Sum, Ex: 371.7 Mean, X: 61.95 Variance, s2: 16.503 Steps N S = i=1 1 (r; - 7), VN 1 s2= Elxi - 7)2 N-1 (55.8 - 61.95)2 + ... + (67.2 - 61.95)2 6-1 82.515 5 = 16.503 s = 116.503 = 4.0623884600072

Further the Calculation of Confidence Interval for population mean is given below,

The formula for 95% Confidence Interval for population mean is,

The provided sample mean is X = 61.95 and the sample standard deviation is s = 4.0624. The size of the sample is n 6 and the required confidence level is 95%. The number of degrees of freedom are df = 6 – 1 = 5, and the significance level is a = - 0.05. Based on the provided information, the critical t-value for a = 0.05 and df = 5 degrees of freedom is te = 2.571. The 95% confidence for the population mean ji is computed using the following expression te xs te XS CI (* Ã + vñ vñ Therefore, based on the information provided, the 95% confidence for the population mean u is < 2.571 x 4.0624 CI = 61.95 2.571 x 4.0624 ,61.95 + VT O = = (61.95 – 4.263, 61.95 + 4.263) + (57.687, 66.213) which completes the calculation.

The t critical value is calculated using t table.

The 95% Confidence Interval for population mean is (57.687, 66.213).

Interpretation : We are 95% Confident that the true population mean height of tomato plants is occupied by this Interval 57.687 & 66.213.

b) similarly the 99% Confidence Interval for population mean is,

The provided sample mean is X = 61.95 and the sample standard deviation is s = = 4.0624. The size of the sample is n = 6 and the required confidence level is 99%. The number of degrees of freedom are df = 6 – 1= 5, and the significance level is a = 0.01. Based on the provided information, the critical t-value for a = = 0.01 and df = 5 degrees of freedom is tc = 4.032. The 99% confidence for the population mean u is computed using the following expression te XS CI (x X ,+ te xs vn na vn Therefore, based on the information provided, the 99 % confidence for the population mean u is < CI = = (61.95 - 4.032 x 4.0624 ,61.95 + Vo 4.032 x 4.0624 Vo - = (61.95 – 6.687,61.95 + 6.687) 0 - (55.263, 68.637) which completes the calculation.

The t critical value is calculated using t table.

The 99% Confidence Interval for population mean is (55.263, 68.637).

Thank you.