Question

The mean weight of all 20-year-old women is 128 pounds. A random sample of 40 vegetarian women ho are 20 years old showed a sample mean of 122 pounds with a standard deviation of 15 pounds. women's measurements were independent of each other. Determine whether the mean weight tor 20 year-old vegetarian women is significantly less than 128, using a significant level of 0.05 The (a) (4 points) State the null and alternative hypotheses using both symbols and words (be sure to use the correct symbols). (b) (3 points) Show that the conditions for the CLT are satisfied. (C) (4 points) Perform the appropriate hypothesis test using technology. Write down what command you used and what numbers you entered. Report the values of the test statistic and p-value. y pothesis? (o d pino Do you rejec (e) (2 points) State your conclusion in the context of this problem.

Answer 1

Solution:

Here, we have to use one sample t test for population mean.

Part a

Null hypothesis: H₀: The mean weight for 20-year old vegetarian women is 128 pounds.

Alternative hypothesis: H_a: The mean weight for 20-year old vegetarian women is significantly less than 128 pounds.

H₀: µ = 128 versus H_a: µ < 128

This is a lower tailed or left tailed (One tailed) test.

Part b

We know that the sampling distribution of the sample mean follows an approximate normal distribution. The population standard deviation is not given, so we use t distribution. Sample size is adequate. Also, all measurements are independent.

Part c

We are given

Level of significance = α = 0.05

Xbar = 122

S = 15

n = 40

df = n – 1 = 40 – 1 = 39

Critical t value = -1.6849

(by using t-table)

Test statistic formula is given as below:

t = (Xbar - µ) / [S/sqrt(n)]

t = (122 – 128) / [15/sqrt(40)]

t = -6/ 2.3717

t = -2.5298

P-value = 0.0078

(by using t-table or excel command =TDIST(ABS(-2.5298),39,1))

Part d

P-value < α = 0.05

So, we reject the null hypothesis

Part e

There is sufficient evidence to conclude that the mean weight for 20-year old vegetarian women is significantly less than 128 pounds.