Question

The president of Amalgamated Retailers International, Sam Peterson, has asked for your assistance in studying the market penetration for the company’s new cell phone. You are asked to determine if the market share is equal to the company’s claim of 35%. You obtain a random sample of potential customers from the area. The sample indicates that 258 out of a total sample of 800 indicate they will purchase from Amalgamated

[a] Using a probability of error , test the hypothesis that the market share equals the company’s claim of 35% versus the hypothesis that the market share is not equal to the company’s claim.

[b] Using a probability of error , test the hypothesis that the market share equals the company’s claim of 35% versus the hypothesis that the market share is less than the company’s claim.

Please Include:

1. The problem statement
2. Assumptions
3. The null and alternate hypothesis statements
4. The significance level
5. The test statistic (as an equation)
6. Decision rules
7. The calculated value of the test statistic
8. The p-value
9. Interpret the results of the test.

Answer 1

n = 800

x = 258

p̄ = x/n = 0.3225

significance level, α = 0.05

Assumption:

• The data are a simple random sample from the population of interest.
• The population is at least 10 times as large as the sample.
• n⋅p≥10 and n⋅(1−p)≥10, where n is the sample size and p is the true population proportion

a) Null and Alternative hypothesis:

Ho : p = 0.35

H1 : p ≠ 0.35

Test statistic:

z = (p̄ -p)/√(p*(1-p)/n) = (0.3225 - 0.35)/√(0.35 * 0.65/800) = -1.6307

Critical value :

Two tailed critical value, z crit = ABS(NORM.S.INV(0.05/2)) = 1.960

Reject Ho if z < -1.96 or if z > 1.96

p-value :

p-value = 2*(1-NORM.S.DIST(ABS(-1.6307), 1)) = 0.1029

Decision:

p-value > α, Do not reject the null hypothesis

Conclusion:

There is not enough evidence to conclude that the market share is not equal to the company’s claim.

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b)

Null and Alternative hypothesis:

Ho : p = 0.35

H1 : p < 0.35

Test statistic:

z = (p̄ -p)/√(p*(1-p)/n) = (0.3225 - 0.35)/√(0.35 * 0.65/800) = -1.6307

Critical value :

Left tailed critical value, z crit = NORM.S.INV(0.05) = -1.645

Reject Ho if z < -1.645

p-value :

p-value = NORM.S.DIST(-1.6307, 1) = 0.0515

Decision:

p-value > α, Do not reject the null hypothesis

Conclusion:

There is not enough evidence to conclude that that the market share is less than the company’s claim.