Question

In the past, patrons at a cinema spent an average of $ 2.50 for popcorn with standard deviation of $ 0.90. The amount of these expenditures is normally distributed . Following an intensive public campaign about the negative health effects of eating popcorn by a local medical school, the mean expenditures for a sample of 18 patrons is found to be $ 2.10. At 0.01 level of significance, does this recent experience suggest decline in spending

Answer 1

To determine if there has been a decline in spending on popcorn at the cinema, we can perform a one-sample t-test. The calculated t-value is compared to the critical value to conclude. In this case, there is sufficient evidence to suggest a decline in spending.

Step-by-step explanation:

To test whether there has been a decline in spending on popcorn at the cinema, we can use a hypothesis test. We will compare the sample mean of the recent expenditures to the population mean of the past expenditures, using the given sample size, standard deviation, and significance level of 0.01.

The null hypothesis (H0) is that there is no decline in spending, meaning the sample mean is equal to the population mean. The alternative hypothesis (Ha) is that there is a decline in spending, meaning the sample mean is less than the population mean.

We can use a one-sample t-test to test this hypothesis. We calculate the test statistic (t) using the formula:

t = (sample mean - population mean) / (standard deviation/sqrt (sample size))

In this case, the sample mean is $2.10, the population mean is $2.50, the standard deviation is $0.90, and the sample size is 18. Plugging these values into the formula, we get:
t = (2.10 - 2.50) / (0.90 / sqrt(18)) = -2.78

We compare the calculated t-value to the critical value from the t-distribution table for a one-tailed test with 17 degrees of freedom (sample size minus 1) and a significance level of 0.01. The critical value is approximately -2.617. Since the calculated t-value is less than the critical value, we reject the null hypothesis and conclude that there is sufficient evidence to suggest a decline in spending on popcorn at the cinema.

Answer 2

There's no evidence to show that the recent experience suggests a decline in spending.

Step-by-step explanation:

We are given;

Population mean; μ = $2.5

Population Standard deviation; σ = $0.9

Sample mean; x¯ = $2.1

Sample size; n = 18

Significance level; α = 0.01

Let's define the hypothesis;

Null hypothesis; H0: μ ≥ $2.5

Alternative hypothesis; Ha: μ < $2.5

z-score formula is;

z = (x¯ - μ)/(σ/√n)

z = (2.1 - 2.5)/(0.9/√18)

z = -0.4/0.2121

z = -1.886

Using the p-value from z-score calculator attached and using z = -1.886; α = 0.01; one tail; we have;

P-value = 0.2965

This is more than the significance level. Thus, we will fail to reject the null hypothesis and conclude that there is no evidence to show that the recent experience suggests a decline in spending.